Discrete Applied Mathematics 162 (2014) 285–295

Contents lists available at ScienceDirect

Discrete Applied Mathematics

journal homepage: www.elsevier.com/locate/dam

(F , I)-security in graphsCaleb PetrieUniversity for Information Science and Technology, ‘‘St. Paul the Apostle’’, Partizanska bb, 6000 Ohrid,Former Yugolav Republic of Macedonia, The

a r t i c l e i n f o

Article history:Received 29 May 2012Received in revised form 28 June 2013Accepted 2 July 2013Available online 20 September 2013

Keywords:SecuritySecurity numberAttackInteger defenseUltra-security

a b s t r a c t

Let G = (V , E) be a graph and S ⊆ V . A set S is (F , I)-secure if every (possibly fractional)attack can be defended by an integer defense. A necessary and sufficient condition for S tobe (F , I)-secure is given. For a graph G, the (F , I)-security number of G is the cardinality ofa smallest (F , I)-secure set of G. The (F , I)-security number for various classes of graphs isdetermined. It is also shown that ultra-security implies (F , I)-security. Some partial resultsand areas for further study are included.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

All graphs in this paper are finite and simple. For a graph G = (V , E) and v ∈ V , we let N(v) = {u ∈ V | uv ∈ E}, andN[v] = N(v) ∪ {v}. For S ⊆ V , N(S) = ∪s∈S N(s) and N[S] = N(S) ∪ S. In this paper, the neighbor set notation N alwaysrefers to the graph G under consideration, and never refers to a subgraph of G. The degree of a vertex v ∈ V will be denotedby degG(v). Most of this paper was first published as part of a doctoral thesis [20], but is significantly revised and expandedin its present form.

Brigham et al. [3] gave a definition of security for nonempty sets S ⊆ V . In [13,20], three new definitions of security weregiven in an effort to explore fractional variants of security. An attack on S is a function A: (V − S) × S → [0, 1] such thatA(u, v) = 0 if uv ∈ E and for u ∈ V − S,

v∈N(u)∩S A(u, v) ≤ 1. A defense of S is a function D: S × S → [0, 1] such that

D(u, v) = 0 if uv ∈ E and u = v, and for u ∈ S,

v∈N[u]∩S D(u, v) ≤ 1. For u ∈ S, let

D∗(u) =

v∈N[u]∩S

D(v, u) and A∗(u) =

v∈N(u)−S

A(v, u).

Attacks and defenses permit real values, but we will still employ the word fractional. This will be explained further after theproof of the Main Theorem. An attack A on S is said to be defendable if there exists a defense D such that D∗(u) ≥ A∗(u) forall u ∈ S. Any such defense is a successful defense of A. An integer attack is an attack whose range is a subset of {0, 1}, and aninteger defense is a defense whose range is a subset of {0, 1}.

In an integer attack, each attacking vertex can either do nothing, or send its entire unit of attack to one of its neighborsin S. Similarly, in an integer defense, each defending vertex either does nothing, or sends its entire unit of defense to anappropriate vertex. The fractional variants allow attackers or defenders (or both) to spread their unit of attack or defenseamong multiple vertices. Security in graphs is a topic that has its roots in alliances in graphs. In [19], there is a list of someareas where alliances and security may be of interest. In some of these cases, it is natural to move beyond an all or nothingapproach. For example, in a war a nation may send its forces in several different directions at one time.

E-mail address: cjp0003@auburn.edu.

0166-218X/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.dam.2013.07.005

286 C. Petrie / Discrete Applied Mathematics 162 (2014) 285–295

The set S is secure if every integer attack on S is defendable by an integer defense. This corresponds to the definitionof Brigham et al. The set S is (I, F)-secure if any integer attack is defendable. Likewise, S is (F , I)-secure if any attack isdefendable by an integer defense, and S is (F , F)-secure if any attack is defendable. Three of the four definitions of securityare equivalent.

Equivalence Theorem ([13,20]). Let G = (V , E) be a graph and S ⊆ V . Then S is secure ⇔ S is (I, F)-secure ⇔ S is (F , F)-secure.

The security number of a graph G, denoted by s(G), is the cardinality of a smallest secure set in G. By the EquivalenceTheorem, a smallest secure set is also a smallest (I, F)-secure set and a smallest (F , F)-secure set. The cardinality of asmallest (F , I)-secure set in G is the (F , I)-security number of G and is denoted by s(F ,I)(G). From the definitions, it is clearthat any (F , I)-secure set is secure, so s(G) ≤ s(F ,I)(G). Brigham et al. show that s(Kn) =

n2

and in [13,20] it is shown that

s(F ,I)(Kn) = n − 1. So security and (F , I)-security are different. Brigham et al. gave the following necessary and sufficientcondition for a set of vertices to be secure.

Theorem BDH ([3]). Let G = (V , E) be a graph and S ⊆ V . The set S is secure if and only if for all X ⊆ S, |N[X]∩S| ≥ |N[X]−S|.

The Main Theorem of this paper gives a necessary and sufficient condition for (F , I)-security.

2. Bipartite graph lemma

We now develop a lemma about bipartite graphs for use in the proof of the Main Theorem. This lemma is also used indetermining the (F , I)-security number of various families of graphs, including complete multipartite graphs. Let A be anattack on S, D a defense of S, and s ∈ S. In the setting of (F , I)-security, D∗(s) is an integer, but A∗(s) might not be an integer.Thus, in order for S to be (F , I)-secure, for any attack A there must be an integer defense D such that D∗(s) ≥ ⌈A∗(s)⌉ for alls ∈ S. In this case,

|S| ≥

s∈S

D∗(s) ≥

s∈S

A∗(s)

. (1)

Given an attack A on a set S in an (F , I)-security setting, the total effective attack is

s∈S ⌈A∗(s)⌉. So Inequality (1) shows thatfor a set S to be (F , I)-secure, |S| must be greater than or equal to the total effective attack of A.

Lemma 2.1 ([20]). Let G be a complete bipartite graph with bipartition (X, Y ). Among (F , I)-attacks on Y , the maximum totaleffective attack achievable is |X | + |Y | − 1.

Proof. Let X = {x1, . . . , xm} and Y = {y1, . . . , yn}. Let A be any attack on Y . Note thatn

j=1 A∗(yj) ≤ m because |X | = m.

We have A∗(yj)

< A∗(yj) + 1, i = 1, . . . , n

⇒

nj=1

A∗(yj)

<

n1

(A∗(yj) + 1) ≤ m + n

⇒

nj=1

A∗(yj)

≤ m + n − 1.

To achieve the bound in this inequality, define an attack A by letting A(x1, yj) =1n for 1 ≤ j ≤ n, and A(xi, y1) = 1 for

2 ≤ i ≤ m. �

The proof of Lemma 2.1 also provides a solution to a proposed problem in [16]. Since any bipartite graph can be viewedas a subgraph of a complete bipartite graph, we have the following corollary:

Corollary 2.1 ([20]). Let G be a bipartite graph with bipartition (X, Y ). For any (F , I)-attack on Y , the total effective attack isless than or equal to |X | + |Y | − 1.

Next we use the Max-Flow Min-Cut Theorem [6,5,1] to show that this upper bound can be achieved when G is a tree.

Lemma 2.2 ([20]). Let G = (V , E) be a tree with bipartition (X, Y ). Define f : V → [0, ∞) by f (v) = 1 for v ∈ X andf (v) = degG(v) −

|Y |−1|Y |

for v ∈ Y . Then there exists a function wt: E → [0, ∞) such that for all v ∈ V ,

f (v) =

e∈E

e incident to v

wt(e).

C. Petrie / Discrete Applied Mathematics 162 (2014) 285–295 287

Proof. Create a digraphD from G by directing every edge from X to Y . Add a vertex s and for each x ∈ X , form an arc directedfrom s to x. Similarly, add a vertex t and for each y ∈ Y , form an arc directed from y to t . Let A[s, X] be the set of arcs withone end at s and one end in X , A[Y , t] be the set of arcs with one end in Y and one at t , and A[X, Y ] denote the arcs withone end in X and one end in Y . From this digraph, form a network by designating s the source and t the sink (see Fig. 1).Define the capacity function c as follows: c(a) = 1 if a ∈ A[s, X], c(a) = degG(y) −

|Y |−1|Y |

if a ∈ A[Y , t], and c(a) = |X | + 1otherwise. Note that

a∈A[Y ,t] c(a) = |X |, because G is a tree.

Fig. 1. The network formed from the bipartite graph G.

If we can find a flow in this network of value |X |, then the desired functionwt on E can be found by assigningwt(e), e ∈ E,to be equal to the flow of the corresponding arc. To show we can find a flow of value |X |, we will show that the minimumcut has capacity |X |. The capacity of a minimum cut is clearly less than or equal to |X |, because we can form a cut bytaking all the arcs of A[s, X] (or all the arcs of A[Y , t]). Now we will show that every other cut has a capacity biggerthan |X |.

Any cut including an arc of A[X, Y ] will have capacity at least |X | + 1. The only other cuts to check must include some,but not all, arcs of A[s, X] and some, but not all, arcs of A[Y , t]. Let K be a cut that has at least one arc of A[s, X] and at leastone arc of A[Y , t]. Let X0 = {x| x ∈ X and (s, x) ∈ K} and Y0 = {y| y ∈ Y and (y, t) ∈ K}. By assumption about K , |X0| ≥ 1and |Y0| ≥ 1. Note that there are no arcs between X0 and Y0; otherwise K would not be a cut.

The capacity of the cut K is

|X − X0| +

y∈Y−Y0

degG(y) −

|Y | − 1|Y |

.

Since each edge of G has one end in X and one end in Y ,

y∈Y−Y0degG(y) is the number of edges of Gwith one end in Y −Y0.

G has |X | + |Y | − 1 edges. Each edge has an end in Y − Y0 or in Y0. An edge with an end in Y0 must have its other end inX − X0, as noted above. Looking at the forest induced in G by (X − X0)∪ Y0, the most edges it can have is |X − X0| + |Y0| − 1.Thus,

y∈Y−Y0

degG(y) ≥ (|X | + |Y | − 1) − (|X − X0| + |Y0| − 1) = |X | − |X − X0| + |Y − Y0|. So we have

c(K) = |X − X0| +

y∈Y−Y0

degG(y) −

|Y | − 1|Y |

≥ |X − X0| + |X | − |X − X0| + |Y − Y0| −

|Y | − 1|Y |

|Y − Y0|

= |X | + |Y − Y0| − |Y − Y0| +|Y − Y0|

|Y |

= |X | +|Y − Y0|

|Y |

> |X |. �

Let G = (V , E) be a connected bipartite graph with bipartition (X, Y ). By Corollary 2.1, |X | + |Y | − 1 is an upper boundfor the total effective attack on Y . Also, G has a spanning tree, and thus can achieve the bound of |X |+ |Y |−1 by Lemma 2.2.The attack that achieves this bound corresponds naturally to the function wt . Thus we have the following lemma.

Bipartite Graph Lemma ([20]). Let G be a connected bipartite graph with bipartition (X, Y ). Among (F , I)-attacks on Y , themaximum total effective attack achievable is |X | + |Y | − 1.

288 C. Petrie / Discrete Applied Mathematics 162 (2014) 285–295

3. Main theorem

Given a graph G = (V , E) and a set S ⊆ V , the set of edges with one end in V − S and one end in S induces a graph whosecomponents are connected bipartite graphs. By looking at the maximum total effective attack achievable by the attackers ineach component, we develop a necessary and sufficient condition for (F , I)-security.

For X ⊆ S, let GX denote the subgraph of Gwhose vertex set is X ∪ (N[X] − S) and whose edge set is E[X,N[X] − S], theset of edges of G with one end in X and the other in N[X] − S. Let c(GX ) denote the number of components of GX . Then letC1, C2, . . . , Cc(GX ) be the components of GX and Xi = V (Ci) ∩ X for 1 ≤ i ≤ c(GX ). In this paper, the neighbor set notation Nalways refers to the graph G, and never to the graph GX .

Each Ci is a connected bipartite graph with bipartition (Xi,N[Xi] − S). By the Bipartite Graph Lemma, the upper boundfor total effective attack from N[Xi] − S to Xi is |Xi| + |N[Xi] − S|−1. Since the sets Xi, 1 ≤ i ≤ c(GX ), are pairwise disjoint,an upper bound for the total effective attack from V − S to X is

c(GX )i=1 (|Xi| + |N[Xi] − S| − 1) = |X | + |N[X] − S| − c(GX ).

By the Bipartite Graph Lemma, for each component Ci, there is an attack that achieves the bound |Xi| + |N[Xi] − S| − 1. Soone component at a time, we can construct an attack from V − S to X achieving the bound |X | + |N[X] − S| − c(GX ).

In order for S to be (F , I)-secure, it is thus necessary that for all X ⊆ S, |N[X] ∩ S| ≥ |X | + |N[X] − S| − c(GX ). The MainTheorem states that this condition is also sufficient:

Main Theorem ([20]). Let G = (V , E) be a graph and S ⊆ V . Then S is (F , I)-secure if, and only if, |N[X] ∩ S| ≥

|X | + |N[X] − S| − c(GX ) for all X ⊆ S.

The use of Theorem HRHV to prove the sufficiency of this condition is similar to the use of Theorem HRHV in [13,21,20].

Theorem HRHV ([9,22,10]). Let P1, P2, . . . , Pn be sets and k1, k2, . . . , kn non-negative integers. There exist pairwise disjointsets D1,D2, . . . ,Dn such that Di ⊆ Pi and |Di| = ki, 1 ≤ i ≤ n, if, and only if, for each I ⊆ {1, . . . , n}, |∪i∈I Pi| ≥

i∈I ki.

Proof of Main Theorem. Necessity has already been proven. Let G = (V , E) be a graph and let S = {s1, s2, . . . , sn} ⊆ V besuch that for all X ⊆ S, |N[X] ∩ S| ≥ |X | + |N[X] − S| − c(GX ). Let A be any attack on S, and Pi = N[si] ∩ S for 1 ≤ i ≤ n. Letki = ⌈A∗(si)⌉ for 1 ≤ i ≤ n. Then Pi is the set of potential defenders of si, and ki is the number of defenders of si needed fora successful defense of A. For any J ⊆ {1, . . . , n}, let XJ =

j∈J{sj}. Then we have

j∈J kj ≤ |XJ | + |N(XJ) − S| − c(GXJ ) ≤

|N[XJ ] ∩ S| = | ∪j∈J(N[sj] ∩ S)| = | ∪j∈J Pj|. By Theorem HRHV we can find pairwise disjoint Di ⊆ Pi, 1 ≤ i ≤ n, such that|Di| = ki for 1 ≤ i ≤ n. Then the defense D defined by

D(si, sj) =

1 if si ∈ Dj0 otherwise

is a successful defense of the attack A. �

The Main Theorem shows that in determining whether or not a set S ⊆ V is (F , I)-secure, we do not have to examineevery possible attack. We only need to consider the maximum total effective attack from N[X] − S to X for each X ⊆ S.In the proof of Lemma 2.2 in Section 2, all of the capacities in the network are rational numbers. Using the Ford–Fulkersonalgorithm [7,1], we can construct a maximum flow that uses only rational values. This maximum flow corresponds to arational valued attack that achieves the maximum total effective attack. Thus, if the definition of an attack is restricted torational values, the maximum total effective attack is the same as when real values are allowed. So while our definition ofa fractional attack allows for real numbers, the sets that are (F , I)-secure would be the same if we restricted our definitionto include only rational numbers.

4. Some (F, I)-security numbers

Recall that for any graph G, s(G) ≤ s(F ,I)(G). Given two graphs G and H , we denote their cartesian product by G�H andtheir join by G∨H . The path on n vertices is denoted by Pn. Let Fn = K1 ∨ Pn andWn = K1 ∨ Cn. As observed in [3], it is clearthat a minimum secure set is connected; that is, a minimum secure set induces a connected subgraph of the graph in whichit is a minimum secure set. Likewise, a minimum (F , I)-secure set is connected. For many of the graphs in Proposition 4.1,the corresponding security number is given in [3]. Similar results on ultra-security, which is defined in Section 5, can befound in [21,20]. Except for part (5), the results of Proposition 4.1 are also in [20].

Proposition 4.1. Let G = (V , E) be a graph.(1) s(F ,I)(G) = 1 if and only if δ(G) ≤ 1.(2) s(F ,I)(G) = 2 if and only if δ(G) = 2 and there exists uv ∈ E such that d(u) = d(v) = 2.(3) s(F ,I)(Pn�Pm) = min{m, n, 4}.(4) min{m, 2n, 6} ≤ s(F ,I)(Cm�Pn) ≤ min{m, 2n, 8}.(5) min{2m, 2n, 12} ≤ s(F ,I)(Cm�Cn) ≤ min{2m, 2n, 16},max{m, n} ≥ 4.(6) s(F ,I)(Fn) = 1 +

n2

, n ≥ 2.

(7) s(F ,I)(Wn) = 1 + n+1

2

, n ≥ 3.

C. Petrie / Discrete Applied Mathematics 162 (2014) 285–295 289

Proof. (1) A vertex of degree zero or one is (F , I)-secure, but any vertex of greater degree is not.(2) By (1), δ(G) > 1 is necessary for s(F ,I)(G) = 2. Let S = {u, v} be (F , I)-secure. Since a minimum (F , I)-secure set isconnected, uv ∈ E. In order to be (F , I)-secure, S must also be secure. So |N[S] − S| ≤ 2, which forces d(u) ≤ 3 andd(v) ≤ 3. If d(u) = 3 or d(v) = 3, then c(GS) = 1, |N[S] − S| = 2, and |S| + |N[S] − S| − c(GS) = 3 > |S|. So by the MainTheorem, S is not (F , I)-secure. Thus d(u) = d(v) = 2 is necessary. If S = {u, v}, uv ∈ E, and d(u) = d(v) = 2, then thedefense where u defends itself and v defends itself is successful against any attack.(3) In [3] it is shown that s(Pn�Pm) = min{m, n, 3}. First we will suppose that min{m, n} ≤ 3. Then s(F ,I)(Pn�Pm) ≥

s(Pn�Pm) = min{m, n, 3} = min{m, n}. The n vertices that make up the end vertices of the Pm paths form an (F , I)-secureset, and them vertices that make up the end vertices of the Pn paths form an (F , I)-secure set. So s(F ,I)(Pn�Pm) = min{m, n}.Now suppose that min{m, n} ≥ 4. Then s(F ,I)(Pn�Pm) ≥ s(Pn�Pm) = min{m, n, 3} = 3. Let S = {s1, s2, s3}, where s2 isa corner vertex with neighbors s1 and s3. Let N[S] − S = {v1, v2, v3} and let {s1s2, s1v1, s1v2, s3s2, s3v2, s3v3} be a subsetof the edges (see Fig. 2). Then the attack A defined by A(v1, s1) = 1, A(v2, s1) = 0.5, A(v2, s3) = 0.5, and A(v3, s3) = 1is not defendable by an integer defense. Any other S such that |S| = 3 satisfies |S| < |N[S] − S| and is not secure, muchless (F , I)-secure. Four vertices that induce a 4-cycle and include a corner vertex forms an (F , I)-secure set. So in this case,s(F ,I)(Pn�Pm) = 4.

Fig. 2. The set S = {s1, s2, s3} is not (F , I)-secure.

(4) Kozawa et al. [18] show that s(Cm�Pn) = min{m, 2n, 6}. This settled a conjecture of Brigham et al. [3]. This gives thelower bound for the (F , I)-security number. Two consecutive copies of Pn form an (F , I)-secure set, as does an m-cycleconsisting of end vertices of the paths. If m ≥ 4 and n ≥ 2, taking the end vertex of a path and its neighbor in the path, infour consecutive paths, gives an (F , I)-secure set of eight vertices.(5) This proof is similar to the proof of (4). Kozawa et al. [18] also show that s(Cm�Cn) = min{2m, 2n, 12} formax{m, n} ≥ 4,proving another conjecture of Brigham et al. [3]. This establishes the lower bound for the (F , I)-security number. Twoconsecutive copies of Cm form an (F , I)-secure set, as do two consecutive copies of Cn. If m, n ≥ 4, then a four by foursquare consisting of sixteen vertices is also (F , I)-secure.(6) Let Fn = K1 ∨ Pn, and suppose S ⊆ V is an (F , I)-secure set such that V (K1) ∈ S. The subgraph GS is connected. By theMain Theorem, it is then necessary that 1 = c(GS) ≥ |N[S] − S|. So S = V (Pn) and |S| = n. Now suppose V (K1) ⊆ S. LetS ′

= V (Pn) ∩ S and note that |S ′| ≤ n − 1 (otherwise S ′

= V (Pn) and |S| = |V (Fn)| = n + 1). Every vertex in V (Pn) − S ′

can attack V (K1) and there is at least one u ∈ V (Pn) − S ′ such that u has a neighbor in S ′. Define an attack by letting u senda half unit of attack to V (K1) and a half unit of attack to its neighbor in S ′. Let every other vertex of V (Pn) − S ′ attack V (K1)with its whole unit of attack. By Inequality (1), we have |S| = 1 + |S ′

| ≥ |V (Pn) − S ′| + 1 = n − |S ′

| + 1. This impliesthat |S ′

| ≥ n

2

and thus |S| ≥ 1 +

n2

. Next we construct an (F , I)-secure set S, such that |S| = 1 +

n2

. Let S be the set

containing V (K1) and the first n

2

vertices of V (Pn). Only two vertices can be attacked: V (K1) and some s ∈ V (Pn) ∩ S. Let s

defend itself, and have all other vertices of S defend the V (K1). Since s has exactly one neighbor in V (Pn)− S, it has sufficientdefense. Likewise, V (K1) has a total defense of

n2

, and the attack there is at most

n2

. This defense is successful against

any attack on S.(7) The proof is similar to the proof of (6). LetWn = K1 ∨ Cn, and suppose S ⊆ V does not include the V (K1). Then V (K1) canattack every vertex in the set S. In order for S to be (F , I)-secure, it can have no other attackers. So S = V (Cn) and |S| = n.Now suppose S ⊆ V includes V (K1). Let S ′

= S∩V (Cn). If |S ′| ≥ n−1, then |S| ≥ n, and we already can find an (F , I)-secure

set of size n. If |S ′| ≤ 1, then |S| ≤ 2 and S is not (F , I)-secure by inspection. (In fact, for n ≥ 4, S is not secure.) So let

2 ≤ |S ′| ≤ n − 2. There are at least three vertices of S that have neighbors outside of S: V (K1) and some u, v ∈ S, u = v.

Let X = {V (K1), u, v}. Then GX is a connected bipartite graph. Since every vertex in V (Cn) − S ′ is adjacent to V (K1), one

290 C. Petrie / Discrete Applied Mathematics 162 (2014) 285–295

part of the bipartition has size |V (Cn) − S ′| = n − |S ′

|. The other part is X and |X | = 3. By the Bipartite Graph Lemma andInequality (1), in order for S to be (F , I)-secure, it is required that |S| = 1 + |S ′

| ≥ n − |S ′| + 3 − 1. Isolating |S ′

| yields|S ′

| ≥ n+1

2

, and thus |S| ≥ 1 +

n+12

.

An (F , I)-secure set of size 1+ n+1

2

can be found by taking V (K1) and

n+12

consecutive vertices of the V (Cn). Exactly

three vertices can be attacked, V (K1), and some u, v ∈ S ′; u and v each have exactly one neighbor in V (Cn) − S ′. Let u andv defend themselves, and have every other vertex of S defend V (K1). Then u and v clearly have sufficient defense, whileV (K1) has a total defense of

n−12

and at most the attack at V (K1) is

n−12

. So this defense is successful against any attack

on S. �

5. Ultra-security implies (F, I)-security

Given a graph G = (V , E), a set S ⊆ V is ultra-secure if there is an integer defense that is a successful defense of anyinteger attack [21,20]. This can be viewed as a situation in which the defenders do not have time to react to the attack, butmust set up their defense before the attack happens. The cardinality of a smallest ultra-secure set in G is the ultra-securitynumber of G, and is denoted by su(G). The following is a necessary and sufficient condition for a set S to be ultra-secure,similar in spirit to Theorem BDH and the Main Theorem.

Theorem 5.1 ([21,20]). Let G = (V , E) be a graph. A set S ⊆ V is ultra-secure if and only if |N[X] ∩ S| ≥

x∈X |N[x] − S| forall X ⊆ S.

Wenowprove a lemma that will be used to show that ultra-security implies (F , I)-security. The idea is that an equivalentformulation of ultra-security requires each vertex of N[S] − S to send one unit of attack along each edge it has into S.

Lemma 5.1 ([20]). Let G = (V , E) be a graph and S ⊆ V . Then S ⊆ V is ultra-secure if and only if there exists an integer defenseD such that for all v ∈ S, D∗(v) ≥ |N[v] − S|.

Proof. Suppose D is an integer defense such that for all v ∈ S, D∗(v) ≥ |N[v]− S|. For any attack A, A∗(v) ≤ |N[v]− S|, andso for all v ∈ V , A∗(v) ≤ |N[v] − S| ≤ D∗(v). Thus D is successful against A, and S is ultra-secure.

If S is ultra-secure, then there exists an integer defense D that is a successful defense of any integer attack on S. For eachv ∈ S, there exists an integer attack such that A∗(v) = |N[v] − S|. Since D is a successful defense of every integer attack,D∗(v) ≥ |N[v] − S|. �

Proposition 5.1 ([20]). Let G = (V , E) be a graph and S ⊆ V . If S is ultra-secure, then S is (F , I)-secure.

Proof. Let S ⊆ V be an ultra-secure set, and A a (possibly fractional) attack on S. Note that for u ∈ N(S) − S and v ∈ S,A(u, v) ≤ 1. So we have A∗(v) =

u∈N(v)−S A(u, v) ≤

u∈N(v)−S 1 = |N[v] − S|. By Lemma 5.1, because S is ultra-secure,

there is an integer defense D such that for all v ∈ S, D∗(v) ≥ |N[v] − S| ≥ A∗(v), and thus S is also (F , I)-secure. �

Note that we now have s(G) ≤ s(F ,I)(G) ≤ su(G) for any graph G. In the proofs regarding the (F , I)-security numbers ofCm�Pn, Cm�Cn, Fn, andWn in Proposition 4.1, it is easy to see that the (F , I)-secure sets constructed are also ultra-secure. Sowe have the following results, which, except for (2), are also in [20].

Proposition 5.2. (1) min{m, 2n, 6} ≤ su(Cm�Pn) ≤ min{m, 2n, 8}.(2) min{2m, 2n, 12} ≤ su(Cm�Cn) ≤ min{2m, 2n, 16},max{m, n} ≥ 4.(3) su(Fn) = 1 +

n2

, n ≥ 2.

(4) su(Wn) = 1 + n+1

2

, n ≥ 3.

6. Complete multipartite graphs

Next we find the (F , I)-security number of complete multipartite graphs. Dutton et al. [4] show that

s(Kn1,n2,...,nk) =

n1 + n2 + · · · + nk

2

(2)

when each vertex has a degree of at least two. In [21,20] it is shown that

su(Kn1,n2) = minn1 + n2 − 2, n1 +

n1(n2 − 1)n1 + 1

, 2 ≤ n1 ≤ n2. (3)

In Section 7, we will use the fact that Eq. (3) implies that su(Kn1,n2) = n1 + n2 − 2 if and only if n2 ≤ 2n1 + 2. In [20] it isshown that for k ≥ 3, 1 ≤ n1 ≤ · · · ≤ nk, and n =

ki=1 ni,

su(Kn1,...,nk) = n −

n

n − nk + 1

. (4)

The proof of Eq. (4) is very similar to the proof of Theorem 6.2, below. We now proceed with a corollary of the BipartiteGraph Lemma.

C. Petrie / Discrete Applied Mathematics 162 (2014) 285–295 291

Corollary 6.1 ([20]). Let G be a complete bipartite graph with bipartition (X, Y ). If S = X ∪Y1, where |Y1| =

|Y |

2

and Y1 ⊆ Y ,

then S is (F , I)-secure.Proof. Let A be an attack on S. Let XA = {x | x ∈ X and A∗(x) > 0}. By applying the Bipartite Graph Lemma to the subgraphinduced by XA ∪ (Y − S), the maximum total effective attack achievable is then

|XA| + |Y − S| − 1 = |XA| +

|Y |

2

− 1.

Construct a defense D as follows. Let every vertex in XA defend itself. This leaves

|Y |

2

− 1 effective attack remaining, but

|Y1| =

|Y |

2

≥

|Y |

2

−1. Since G is complete, the vertices of Y1 can send their units of defense wherever they are necessary

to finish constructing D. �

We will employ Corollary 6.1 in the proofs of Theorems 6.1 and 6.2, below. By Proposition 4.1 part (2), s(F ,I)(K2,2) = 2.

Theorem 6.1 ([20]). If 2 ≤ n1 ≤ n2 and n2 = 2, then s(F ,I)(Kn1,n2) = n1 + n2

2

.

Proof. Let the bipartition of Kn1,n2 be (X, Y ) where |X | = n1 and |Y | = n2, and let S ⊆ X ∪ Y be a minimum (F , I)-secureset. Note that 2 ≤ |S| ≤ n1 + n2 − 1, and since S is connected, S ∩ Y = ∅ and S ∩ X = ∅. The set S must therefore containvertices of both X and Y . Let x = |X ∩ S| and y = |Y ∩ S|, so that x+ y = |S|. Also note that x ≥ 1 and y ≥ 1. There are threepossible scenarios:(1) 1 ≤ x < |X | and 1 ≤ y < |Y |. In this case, (X ∩ S) ∪ (Y − S) induces a complete bipartite graph, so by the Bipartite

Graph Lemma there exists an attack from Y − S to X ∩ S with total effective attack |Y − S|+ |X ∩ S|−1 = n2 −y+ x−1.Likewise, (Y ∩ S) ∪ (X − S) induces a complete bipartite graph, so there exists an attack from X − S to Y ∩ S with totaleffective attack |X − S| + |Y ∩ S| − 1 = n1 − x + y − 1. Since these two complete bipartite graphs are disjoint, thereexists an attack from V (Kn1,n2) − S to S with total effective attack (n2 − y + x − 1) + (n1 − x + y − 1) = n1 + n2 − 2.Inequality (1) requires that |S| ≥ n1 + n2 − 2. Taking x = |X | − 1 = n1 − 1 and y = |Y | − 1 = n2 − 1 provides an(F , I)-secure set of this size. The integer defense where each vertex of S defends itself is successful against any attack.

(2) x = |X | and 1 ≤ y < |Y |. Only vertices in X can be attacked. There is a complete bipartite graph induced by X ∪ (Y − S).So there is an attack from Y − S to X with total effective attack |X | + |Y − S| − 1 = x + n2 − y − 1 = n1 + n2 − y − 1.By Inequality (1), in order for S to be (F , I)-secure, it is necessary that |S| = x + y = n1 + y ≥ n1 + n2 − y − 1. Thusy ≥

n2−1

2

=

n22

. So |S| = x + y ≥ n1 +

n22

. On the other hand, any set S with all the vertices of X and

n22

vertices of Y is (F , I)-secure, by Corollary 6.1.

(3) 1 ≤ x < |X | and y = |Y |. Similar to case (2), the smallest (F , I)-secure set in this case is of size n2 + n1

2

.

We have s(F ,I)(Kn1,n2) = min{n1 + n2 − 2, n1 + n2

2

, n2 +

n12

}. The minimum is n1 +

n22

except when n1 = n2 = 2.

So if n2 = 2, s(F ,I)(Kn1,n2) = n1 + n2

2

. �

Theorem 6.2 ([20]). Let k ≥ 3, 1 ≤ n1 ≤ n2 ≤ · · · ≤ nk, and n =k

j=1 nj. Then s(F ,I)(Kn1,n2,...,nk) = n − nk

2

.

Proof. Let {X1, X2, . . . , Xk} be the partition of Kn1,n2,...,nk so that |Xi| = ni for 1 ≤ i ≤ k. Let V = V (Kn1,n2,...,nk) andS ⊆ V be a minimum (F , I)-secure set. Note that since |S| ≥ 2 and S is connected, |{i : S ∩ Xi = ∅}| ≥ 2. If|{i : (V − S) ∩ Xi = ∅}| ≥ 2, then the graph induced by the edges with one end in S and one end in V − S induces aconnected bipartite graphwith bipartition (S, V −S). By the Bipartite Graph Lemma, V −S can attack S with a total effectiveattack of |V − S| + |S| − 1 ≥ 2 + |S| − 1 = |S| + 1. By Inequality (1), S cannot be (F , I)-secure. Thus V − S = ∅ or|{i : (V − S) ∩ Xi = ∅}| = 1. If V − S = ∅, then S = V . This is not a smallest (F , I)-secure set, because any set S with|S| = |V | − 1 is (F , I)-secure.

So we must have |{i : (V − S) ∩ Xi = ∅}| = 1. Let Xα be such that (V − S) ∩ Xα = ∅ and (V − S) ∩ Xj = ∅ for all j = α.Let x = |S ∩ Xα|. Then |V − S| = |(V − S) ∩ Xα| = nα − x. There are no edges between S ∩ Xα and V − S because both arecontained in Xα . So the complete bipartite graph induced by the edges with one end in V −S and one end in S has bipartition(V −S, S−Xα). There is an attack from V −S to S−Xα with total effective attack |V −S|+|S−Xα|−1 = nα −x+|S|−x−1.By Inequality (1), it is necessary that |S| ≥ nα − 2x − 1 + |S|, from which it follows that x ≥

nα−12

=

nα

2

. We can find

an (F , I)-secure set of size

j=α nj + nα

2

= n −

nα

2

by choosing any

nα

2

vertices of Xα along with ∪j=α Xj. This set S

is (F , I)-secure by Corollary 6.1, because S is (F , I)-secure in the complete bipartite graph induced by edges with one end inXα and one end in V −Xα . The other edges in the multipartite graph do not allow for any new attack possibilities. Therefore,s(F ,I)(Kn1,n2,...,nk) = min1≤j≤k

n −

nj2

= n −

nk2

. �

7. Open problems

What follows are some classes of graphs for which the (F , I)-security number and ultra-security number are unknown,as well as some other areas for further exploration. For a list of problems involving only the original definition of security,see [3]. Many of the problems from this list also remain open.

292 C. Petrie / Discrete Applied Mathematics 162 (2014) 285–295

7.1. Cm�Pn and Cm�Cn

Find the (F , I)-security number and ultra-security number for Cm�Pn and Cm�Cn. My conjecture is that s(F ,I)(Cm�Pn) =

su(Cm�Pn) = min{m, 2n, 8} and s(F ,I)(Cm�Cn) = su(Cm�Cn) = min{2m, 2n, 16}. The corresponding results for s(Cm�Pn)and s(Cm�Cn) were proved by Kozawa et al. [18].

7.2. The n-cube

The n-cube, denoted by Qn, can be defined inductively as follows: Let Q0 ∼= K1; then Qn−1 = Qn�K2. Note that Qn is ann-regular bipartite graph with |V (Qn)| = 2n and |E(Qn)| = n2n−1. It is clear that s(Q0) = s(F ,I)(Q0) = su(Q0) = s(Q1) =

s(F ,I)(Q1) = su(Q1) = 1, and s(Q2) = s(F ,I)(Q2) = su(Q2) = 2. If the number of vertices of a subgraph of Qn is known,Graham’s Density Lemma gives an upper bound on the number of edges the subgraph may contain.

Graham’s Density Lemma ([8,11,2]). Let H be a subgraph of Qn. Then |E(H)| ≤12 |V (H)| log2 |V (H)|.

Can the following bounds for s(Qn), s(F ,I)(Qn), and su(Qn) be improved?

Proposition 7.1 ([20]). For n ≥ 1, 2⌊n2⌋ ≤ s(Qn) ≤ s(F ,I)(Qn) ≤ su(Qn) ≤ 2n−1.

Proof. Let S ⊆ V (Qn) be secure. By Theorem BDH, for each v ∈ S, |N[v] ∩ S| ≥ |N[v] − S|. It follows that 2|N[v] ∩ S| ≥

|N[v] ∩ S| + |N[v] − S| = |N[v]|, and thus |N[v] ∩ S| ≥12 |N[v]| =

12 (n+ 1). Now let H be the subgraph of Qn induced by S.

Then we have 1 + degH(v) = |N[v] ∩ S| ≥12 (n + 1) and degH(v) ≥

n−12

=

n2

. So H must have at least 1

2 |S| n

2

edges.

Applying Graham’s Density Lemma, we have

12|S|

n2

≤ |E(H)| ≤

12|S| log2 |S|,

which yields 2⌊n2⌋ ≤ |S|.

Now let S ⊆ V such that the subgraph induced by S is isomorphic to Qn−1. Then |S| = 2n−1 and S is ultra-securebecause for all s ∈ S, |N[s] − S| = 1. So the defense where every s ∈ S defends itself is successful against any attack. Thussu(Qn) ≤ 2n−1. �

7.3. The Kneser graph

Let m ≥ k ≥ 1. The Kneser Graph, K(m, k), hasm

k

vertices, such that each vertex is a unique k-subset of a given

m-set. Two subsets are adjacent if and only if they are disjoint. This is the notation used in [4], where it is shown thats(K(m, 2)) =

n+12

for m ≥ 6. What is s(F,I)(K(m, 2)) and su(K(m, 2))? The values of s(K(m, k)), s(F ,I)(K(m, k)) and

su(K(m, 2)) for k ≥ 3 are also open.

7.4. Relationships between security, (F , I)-security, and ultra-security

Let G = (V , E) be a graph, and S ⊆ V be secure.What additional conditions (other than theMain Theorem) are necessaryand sufficient to show that S is also (F , I)-secure? If for all v ∈ N[S]− S, |N[v]∩ S| ≤ 1, then each attacking vertex v cannotsplit its attack amongmultiple vertices. A defense where each vertex of S defends itself is successful against any attack. So Swill be ultra-secure (and thus (F , I)-secure also). On the other hand, the graph K3 shows that this condition is not necessary.For K3, if |S| = 2, then S is secure, (F , I)-secure, and ultra-secure. Yet for v ∈ N[S] − S, |N[v] ∩ S| = 2. The K3 examplecan be generalized to an infinite family of graphs. Proposition 7.2, below, is slightly expanded from its original presentationin [20].

Proposition 7.2. For each positive integer n ≥ 2, there exists a graph G = (V , E) such that s(G) = s(F ,I)(G) = su(G) = n.Moreover, there is a set S ⊆ V that is a minimum ultra-secure set, a minimum (F , I)-secure set, and a minimum secure set, forwhich there exists v ∈ N[S] − S such that |N[v] ∩ S| = 2.

Proof. We saw above that K3 is such a graph for n = 2. For n = 3, the graph G = (V , E) in Fig. 3 (on the following page)satisfies s(G) = s(F ,I)(G) = su(G) = 3. The set S = {s1, s2, s3} is an ultra-secure set of order three; the defense where eachvertex of S defends itself is successful against any attack. Clearly s(G) = 1 because δ(G) > 1, and s(G) = 2 because anyS ⊆ V such that |S| = 2 satisfies |N[S] − S| ≥ 3 (see [3] for the necessary and sufficient conditions for a graph to havesecurity numbers one, two, or three). So s(G) = 3, and thus s(F ,I)(G) = su(G) = 3 also.

For n ≥ 4, we generalize the graph in the figure. Let G = (V , E) be such that |V | = 3n + 1. Let the vertices of the graphbe {s1, . . . , sn} ∪ {v1, . . . , v2n+1}. Let E = {sisi+1|1 ≤ i ≤ n − 1} ∪ {vjvk|j = k} ∪ {visi|1 ≤ i ≤ n − 1} ∪ {v1sn}. Then{v1, . . . , v2n+1} induces a K2n+1 and {s1, . . . , sn} induces a Pn. Letting each si defend itself shows that S = {s1, . . . , sn} isultra-secure of size n. So we have su(G) ≤ n. Also, the vertex v1 is an element of N[S] − S and |N[v1] ∩ S| = |{s1, sn}| = 2.

C. Petrie / Discrete Applied Mathematics 162 (2014) 285–295 293

Fig. 3. A graph such that s(G) = s(F ,I)(G) = su(G) = 3.

We will now show s(G) ≥ n, and the result follows. Let S ⊆ V be secure. If vj ∈ S for some j, by Theorem BDH we have2|N[vj] ∩ S| ≥ |N[vj] ∩ S| + |N[vj] − S| = |N[vj]| ≥ 2n + 1, and thus |S| ≥ n + 1. So no minimum secure set containsvj, j ∈ {1, . . . , 2n + 1}. So if there is a secure set with less than n elements, it has to be a proper subset of {s1, . . . , sn}. Wehave stated previously that a minimum secure set is connected. So let T be a subset of {s1, . . . , sn} such that |T | ≤ n − 1,and T induces a connected subgraph. Then |(N[T ]− T )∩ {v1, . . . , v2n+1}| = |T | and |(N[T ]− T )∩ {s1, . . . , sn}| ≥ 1, so that|T | < |T | + 1 ≤ |N[T ] − T |, and T is not secure. �

The family of graphs in Proposition 7.2 is an example of a graph G such that s(G) = s(F ,I)(G) = su(G). Any graph Gwith s(G) = 1 also satisfies s(G) = s(F ,I)(G) = su(G). The complete graph Kn, for n ≥ 2 satisfies s(Kn) =

n2

[3],

s(F ,I)(Kn) = n − 1 [13,20], and su(Kn) = n − 1 [21,20]. So for n ≥ 4, Kn is an example of a graph G such thats(G) < s(F ,I)(G) = su(G).

Next, we look at the complete bipartite graph Kn,n, for n ≥ 3. By Eq. (2) we have s(Kn,n) = n, by Theorem 6.1 we haves(F ,I)(Kn,n) = n +

n2

, and by Eq. (3) we have su(Kn,n) = 2n − 2. So for n ≥ 5, s(Kn,n) < s(F ,I)(Kn,n) < su(Kn,n) [20].

The following is an example of a graph G such that s(G) = s(F ,I)(G) < su(G).

Example 7.1 ([20]). We will show the graph G in Fig. 4 satisfies s(G) = s(F ,I)(G) = 3 and su(G) = 4. There is no vertex ofdegree one, and no subset of size two is secure, so s(G) ≥ 3. Applying theMain Theorem shows that {a, b, f } is (F , I)-secure,so s(G) = s(F ,I)(G) = 3. As with security and (F , I)-security, a minimum ultra-secure set is also connected. No subset of sizethree is ultra-secure: there are 14 subsets of size three that induce a connected subgraph in G, and each fails the necessaryand sufficient condition for ultra-security given in Theorem 5.1. The set {b, c, e, f } is ultra secure. The defense where eachvertex defends itself is a successful defense of any attack. �

Fig. 4. A graph G such that s(G) = s(F ,I)(G) < su(G).

Now we look at one condition which will guarantee an (F , I)-secure set is also ultra-secure. Recall that given a graphG = (V , E) and X ⊆ S ⊆ V , we define GX to be the subgraph of Gwhose vertex set is X ∪ (N[X] − S) and whose edge set isthe set of all edges of Gwith one end in X and one end in N[X] − S.

294 C. Petrie / Discrete Applied Mathematics 162 (2014) 285–295

Proposition 7.3 ([20]). Let G = (V , E) be a graph, and S ⊆ V . If S is (F , I)-secure and GS is a forest, then S is ultra-secure.Proof. Suppose S is (F , I)-secure and that GS is a forest. Then for X ⊆ S, GX is a forest because E(GX ) ⊆ E(GS). Since GX

is a forest |E(GX )| = |V (GX )| − c(GX ) = |X | + |N[X] − S| − c(GX ). On the other hand, recalling that GX is bipartite withbipartition (X,N[X] − S), we have |E(GX )| =

x∈X |N[x] − S|. Since S is (F , I)-secure, by the Main Theorem we have, for

all X ⊆ S, |N[X] ∩ S| ≥ |X | + |N[X] − S| − c(GX ) = |E(GX )| =

x∈X |N[x] − S|. So, by Theorem 5.1, S is ultra-secure. �

There are many questions that could be explored from these initial results, such as: For each n ≥ 4, does there exista graph G satisfying n = s(G) = s(F ,I)(G) < su(G)? If n = s(G) = s(F ,I)(G), how large can su(G) − n be? Other thanTheorem 5.1, what are the necessary and sufficient conditions for an (F , I)-secure set to be ultra-secure? Can we classify allgraphs G satisfying s(G) = s(F ,I)(G) = su(G)?

7.5. Weighted vertices

Alliances in graphs with weighted vertices have been studied [14,15,17,12]. What follows is one possible definition forsecurity with weighted vertices, similar to the definition of a weighted alliance given in [14]. Let G = (V , E) be a graph andS ⊆ V . Let N denote the set of nonnegative integers. A weighting of G is a function w: V → N. A weighted attack on S is afunction A: (V − S) × S → N such that A(u, v) = 0 if uv ∈ E and for u ∈ V − S,

v∈N(u)∩S A(u, v) ≤ w(u). A weighted

defense of S is a function D: S × S → N such that D(u, v) = 0 if u = v and uv ∈ E, and for u ∈ S,

v∈N[u]∩S D(u, v) ≤ w(u).For u ∈ S, D∗(s) and A∗(s) are defined as before. A weighted attack is defendable if there exists a weighted defense D such

that for each u ∈ S, D∗(u) ≥ A∗(u). A set S ⊆ V is w-secure if every weighted attack for a given w is defendable. We canview each vertex v ∈ N[S] as havingw(v) units of attack (or defense), which it can use one at a time. The following theoremagain uses Theorem HRHV to give a necessary and sufficient condition for a set S to be w-secure. The first part of the proofis similar to a lemma found in [13,20].

Theorem 7.1. Let G = (V , E) be a graph, w a weighting of G, and S ⊆ V . Then S is w-secure if, and only if, for all X ⊆ S,u∈N[X]−S

w(u) ≤

v∈N[X]∩S

w(v). (5)

Proof. Suppose there exists X ⊆ S, such that Inequality (5) fails. Let A be an attack such that for u ∈ N[X] − S,x∈X A(u, x) = w(u). Let D be any defense of S. Then

x∈X

D∗(x) =

x∈X

v∈N[x]∩S

D(v, x) =

v∈N[X]∩S

x∈X

D(v, x)

≤

v∈N[X]∩S

w(v) <

u∈N[X]−S

w(u) =

u∈N[X]−S

x∈X

A(u, x)

=

x∈X

u∈N[x]−S

A(u, x) =

x∈X

A∗(x).

So there must be an x ∈ X satisfying D∗(x) < A∗(x). So S is not w-secure.Now suppose S = {s1, . . . , sn} is a set such that for all X ⊆ S, Inequality (5) holds. Let A be an attack. For 1 ≤ i ≤ n, let

ki = A∗(si). For 1 ≤ i ≤ n, if w(si) > 0, define Si = {si1, si2, . . . , siw(si)}. If w(si) = 0, let Si = ∅. The set Si represents thedifferent units that the vertex si can use in defense. For 1 ≤ i ≤ n, let Li = {i} ∪ {j|sisj ∈ E}. Let Pi =

l∈Li

Sl. The set Pi is theset of potential defense units for vertex si.

Let J ⊆ {1, . . . , n}, XJ = {sj|j ∈ J}, and NJ = {i | si ∈ N[XJ ] ∩ S}. We have

j∈J kj =

j∈J A∗(sj) ≤

u∈N[XJ ]−S w(u) ≤

v∈N[XJ ]∩S w(v) =

i∈NJ|Si| =

∪j∈J Pj .

By Theorem HRHV we can find pairwise disjoint D1, . . . ,Dn such that Di ⊆ Pi and |Di| = A∗(si) for 1 ≤ i ≤ n. We canthen construct the defense function D by defining D(si, sj) = |Si ∩ Dj| for si, sj ∈ S. This satisfies the definition for a defensebecause for si ∈ S,

v∈N[si]∩S D(si, v) =

l∈Li

D(si, sl) =

l∈Li|Dl ∩ Si| ≤ |Si| = w(si). Also, if i = j and sisj ∈ E, then

D(si, sj) = |Si ∩ Dj| = 0. Since D∗(si) = A∗(si) for 1 ≤ i ≤ n, D is a successful defense of A. �

There are many other potential definitions of security in the context of weighted vertices. Combining the ideas offractional attacks and fractional defenses with weighted vertices is one further possibility. Equivalences between variousdefinitions could be explored. Is it possible to model some of these weighted versions of security on graphs without usingweights? Can we replace each vertex v by an unweighted Kw(v) and appropriate edges?

Acknowledgments

The author is indebted to his advisor, Dr. Peter Johnson, for his help in researching and writing this paper. Dr. Garth Isaaksuggested a connection between theMax-FlowMin-Cut Theorem and security. The comments, corrections, and suggestionsof the reviewers significantly improved this paper. The author also thanks Dr. Dorisa Costello for proofreading the use of theEnglish language in a draft of this paper.

C. Petrie / Discrete Applied Mathematics 162 (2014) 285–295 295

References

[1] J.A. Bondy, U.S.R. Murty, Graph Theory, in: Graduate Texts in Mathematics, Springer, 2008.[2] B. Brešar, W. Imrich, S. Klavžar, B. Zmazek, Hypercubes as direct products, SIAM J. Discrete Math. 18 (2005) 778–786.[3] R.C. Brigham, R.D. Dutton, S.T. Hedetniemi, Security in graphs, Discrete Appl. Math. 155 (2007) 1708–1714.[4] R.D. Dutton, R. Lee, R.C. Brigham, Bounds on a graph’s security number, Discrete Appl. Math. 156 (2008) 695–704.[5] P. Elias, A. Feinstein, C.E. Shannon, A note on the maximum flow through a network, IRE. Trans. Inf. Theory 2 (4) (1956) 117–119.[6] L.R. Ford Jr., D.R. Fulkerson, Maximal flow through a network, Canad. J. Math. 8 (1956) 399–404.[7] L.R. Ford Jr., D.R. Fulkerson, A simple algorithm for finding maximal network flows and an application to the Hitchcock problem, Canad. J. Math. 9

(1957) 210–218.[8] R.L. Graham, On primitive graphs and optimal vertex assignments, Ann. New York Acad. Sci. 75 (1970) 170–186.[9] P. Hall, On representatives of subsets, J. Lond. Math. Soc. 10 (1935) 26–30.

[10] P.R. Halmos, H.E. Vaughan, The marriage problem, Amer. J. Math. 72 (1950) 214–215.[11] R. Hammack, W. Imrich, S. Klavžar, Handbook of Product Graphs, 2nd edition, CRC Press, 2011.[12] A. Harutyunyan, A fast algorithm for powerful alliances in trees, in: Combinatorial Optimization and Applications, Springer, Berlin, Heidelberg, 2010,

pp. 31–40.[13] G. Isaak, P. Johnson, C. Petrie, Integer and fractional security in graphs, Discrete Appl. Math. 160 (2012) 2060–2062.[14] L. Jamieson, Algorithms and complexity for alliances and weighted alliances of various types, Ph.D. thesis, Clemson University, 2007. Available at

http://etd.lib.clemson.edu/documents/1181327200 (Last accessed 6.03.2013).[15] L. Jamieson, B.C. Dean, Weighted alliances in graphs, Congr. Numer. 187 (2007) 76–82.[16] P.D. Johnson Jr., C. Petrie, Problems and solutions: problem8, Alabama J.Math. 35 (2010) Spring/Fall. Available at http://ajmonline.org/2010/problems.

pdf (Last accessed 6.03.2013).[17] K. Kimura, M. Koyama, A. Saito, Small alliances in a weighted graph, Discrete Appl. Math. 158 (2010) 2071–2074.[18] K. Kozawa, Y. Otachi, K. Yamazaki, Security number of grid-like graphs, Discrete Appl. Math. 157 (2009) 2555–2561.[19] P. Kristiansen, S.M. Hedetniemi, S.T. Hedetniemi, Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004) 157–177.[20] C. Petrie, Security, (F , I)-security, and ultra-security in graphs, Ph.D. thesis, Auburn University, 2012. Available at http://etd.auburn.edu/etd/handle/

10415/3306.[21] C. Petrie, Ultra-security in graphs, Congr. Numer. 214 (2012) 75–80.[22] R. Rado, A theorem on general measure functions, Proc. Lond. Math. Soc. 44 (1938) 61–91.