On the geodesic Carathéodory number for cartesian product ...liliana/lawclique_2016/lawcg.pdf ·...

On the geodesic Caratheodory number forcartesian product of graphs

Diane Castonguay Eduardo S. LiraErika M. M. Coelho Hebert Coelho da Silva

Universidade Federal de Goias, Brasil

Contents

I DefinitionsI Convexity and Convex SetsI Geodesic ConvexityI Caratheodory NumberI Cartesian Product

I The Caratheodory number for cartesian products

I Conclusion and Questions

Graph Convexity

I Given a graph G and a set C of subsets of V(G)

I C is a convexity on V(G) if

I ∅,V(G) ∈ CI C is closed under intersection

Graph Convexity

I Given a graph G and a set C of subsets of V(G)

I C is a convexity on V(G) ifI ∅,V(G) ∈ CI C is closed under intersection

Convex Hull

I Every member of C is a convex set

I The convex hull of S, denoted HC(S), is the smallest convexset containing S

Convex Hull

I Every member of C is a convex set

I The convex hull of S, denoted HC(S), is the smallest convexset containing S

Graph convexities defined by paths

I Let P be a family of paths of a graph G

I S belongs to CP if every path of P, joining two vertices of S,is contained in S

Graph convexities defined by paths

I Let P be a family of paths of a graph G

I S belongs to CP if every path of P, joining two vertices of S,is contained in S

Examples

I Shortest paths - Geodesic convexity

I Induced paths - Monophonic convexityI Triangle path - Triangle path convexity

I Triangle path is a path which allows only short chords.

I Paths of length two: for multipartite tournaments - p3

convexity

I Paths of length two: in general undirected graphs - p3

convexity

Examples

I Induced paths - Monophonic convexity

I Triangle path - Triangle path convexity

convexity

Examples

convexity

Examples

convexity

Examples

convexity

Geodesic Convexity

The Caratheodory Number

What is the minimum of F such that v ∈ H(F), forall v ∈ H(F) \ F?

I ve ∈ H({va, vc})

I vb ∈ H({va, vc})

I vf ∈ H({va, vc})

I vh ∈ H({va, vd})

I vg ∈ H({va, vc, vd})

I So, given G and S = {va, vc, vd}, |F| = 3.

Definition

Caratheodory numberLet G be a graph. The Caratheodory number, c(G), is the smallestnumber c such that for every S ⊆ V(G) and p ∈ HC[S], thereexists F ⊆ S with |F| 6 c such that p ∈ HC[F]

Problem: Caratheodory numberInstance: A graph G and a integer k.Question: Is the Caratheodory number of G at least k?

Definition

Caratheodory numberLet G be a graph. The Caratheodory number, c(G), is the smallestnumber c such that for every S ⊆ V(G) and p ∈ HC[S], thereexists F ⊆ S with |F| 6 c such that p ∈ HC[F]

Problem: Caratheodory numberInstance: A graph G and a integer k.Question: Is the Caratheodory number of G at least k?

Caratheodory set

I Define ∂HC(S) as HC(S) \⋃

u∈SHC(S \ {u})

I A set S ⊆ V(G) is a Caratheodory set of C if ∂HC(S) is notempty

I Caratheodory number of C is the largest cardinality of aCaratheodory set of C.

Caratheodory set

u∈SHC(S \ {u})

Caratheodory set

u∈SHC(S \ {u})

Caratheodory set

Definition

Cartesian ProductThe cartesian product of two graphs G and H is the graph G�H,with

V(G�H) = {(g,h)|g ∈ V(G) and h ∈ V(H)},

E(G�H) = {(g,h)(g ′,h ′)|g = g ′, hh ′ ∈ E(H), or gg ′ ∈E(G), h = h ′}.

Definition

Cartesian ProductThe cartesian product of two graphs G and H is the graph G�H,with

V(G�H) = {(g,h)|g ∈ V(G) and h ∈ V(H)},

E(G�H) = {(g,h)(g ′,h ′)|g = g ′, hh ′ ∈ E(H), or gg ′ ∈E(G), h = h ′}.

Known results for the Caratheodory number onP3-convexity

I c(G) = 2, if G is a cycle, a path, or a complete graph.

I c(G) = 2, if G is a chordal graph.

I c(G) = 3, if G is a split graph.

I c(G) is unlimited for trees and block graphs

I c(G) 6 n(G)+12

, for general graphs

I c(G) 6 817n(G) + 12

17, for claw-free graphs.

I Caratheodory number is NP-complete for bipartite graphs.

ReferenceR. M. Barbosa, E. M.M. Coelho, M. C. Dourado, D. Rautenbach, and J. L.Szwarcfiter. On the Caratheodory Number for the Convexity of Paths of OrderThree, SIAM J. Discrete Math., 26(3), 929 939, 2012.

I c(GG) 6 3, if G is a chordal graph.

I Caratheodory number is NP-complete for Complementary prisms.

ReferenceM. A. Duarte; L. Penso; D. Rautenbach; U. dos Santos Souza, Complexityproperties of complementary prisms. Journal of Combinatorial Optimization,1–8, 2015.

Known results for the Caratheodory number on otherconvexities

I Monophonic ⇒ c(G) = 1, for complet graphs, c(G) = 2, for other graphs(Duchet - 1988)

I Triangle path ⇒ c(G) = 2 (Changat and Mathew - 1999)

Known results for the Caratheodory number on geodesicconvexity

I NP-complete

I Bipartite graphs ⇒ NP-complete

I Split graph ⇒ c(G) 6 3

ReferenceM.C. Dourado, D. Rautenbach, V.F. dos Santos, P.M. Schafer, and J.L.Szwarcfiter, On the Caratheodory number of interval and graph convexities,Theoretical Computer Science, 2013, 127–135.

Motivation

I The Caratheodory number is unlimited.

ReferenceD. Castonguay, E.M.M. Coelho, E.S. Coelho, E.S. Lira, The geodeticCaratheodory Number, I ETC, 2016.

MotivationI The Caratheodory number is unlimited.

ReferenceD. Castonguay, E.M.M. Coelho, E.S. Coelho, E.S. Lira, The geodeticCaratheodory Number, I ETC, 2016.

Motivation

Is the Caratheodory number still unlimited even when restricted tocartesian products?

Can we determine the Caratheodory number for some graph cartesianproducts?

Motivation

Is the Caratheodory number still unlimited even when restricted tocartesian products?

Can we determine the Caratheodory number for some graph cartesianproducts?

The Caratheodory number for cartesian products

TheoremLet Pn and Pm be the paths of order n and m, respectively. Then,c(Pn�Pm) 6 2.

Sketch of proof.

TheoremLet Kn and Km be the complete graphs of order n and m, respectively. Then,c(Kn�Km) 6 2.

TheoremLet Pn be the path of order n and Km be the complete graph of order m.Then, C(Pn�Km) 6 2.

LemmaLet L be a graph, S ⊆ V(L) and vi, vj two vertices of S. Let G be the cartesianproduct L�K2. No minimum path between vi and vj is introduced in G, andtherefore, H(S) ⊆ V(L) in G.

Sketch of proof.

We now revisit the construction of a family of graphs from which every graphGi was shown by Castonguay et. al to have a Caratheodory set of cardinality i.

The Caratheodory number for cartesian productsG1:

The Caratheodory number for cartesian productsGi+1:

TheoremEvery Gi�K2, where Gi is builty as seen before, has a Caratheodory number ofcardinality i.

Sketch of proof.

CorolaryThe Caratheodory number is unlimited on geodesic convexity, even whenrestricted to the cartesian product.

ProofGiven i, one can construct the graph Gi�K2, with c(Gi�K2) > i.

CorolaryThe Caratheodory number is unlimited on geodesic convexity, even whenrestricted to the cartesian product.

ProofGiven i, one can construct the graph Gi�K2, with c(Gi�K2) > i.

THANK YOU.

Obrigada.

Bibliography

Caratheodory, C., Uber den Variabilitatsbereich der FourierschenKonstanten von positiven harmonischen Funktionen, Rend. Circ. Mat.Palermo, 32 (1911), 193–217.

Changat, M., and J. Mathew, On triangle path convexity in graphs,Discrete Math, 206 (1999), 91–95.

Coelho, E.M., M.C. Dourado, D. Rautenbach, and J.L. Szwarcfiter, TheCaratheodory number of the P3 convexity of chordal graphs, The SeventhEuropean Conference on Combinatorics, Graph Theory and Applications,16 (2013), 209–214.

Bibliography

Dourado, M.C., D. Rautenbach, V.F. dos Santos, P.M. Schafer, and J.L.Szwarcfiter, On the Caratheodory number of interval and graphconvexities, Theoretical Computer Science, 510 (2013), 127–135.

Duchet, P., Convex sets in graphs II. Minimal path convexity, J. CombinTheory, Ser. B, 44 (1988), 307–316.

Parker, D.B, R.F Westhoff, and M.J. Wolf, On two-path convexity inmultipartite tournaments, European J. Combin., 29 (2008), 641–651.

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