Edge-connectivity of permutation hypergraphs Zolt an Szigeti - PowerPoint PPT Presentation

Edge-connectivity of permutation hypergraphs Zolt´ an Szigeti Laboratoire G-SCOP INP Grenoble, France 29 september 2010 joint work with Neil Jami, Ensimag, INP Grenoble, France Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 1 / 1

Outline 1 Permutation graphs 2 Splitting off in graphs 3 Permutation hypergraphs 4 Splitting off in hypergraphs Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 2 / 1

Permutation graphs Definition Given a graph G on n vertices and a permutation π of [ n ], we define the permutation graph G π as follows : 1 we take 2 disjoint copies G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) of G , 2 for every vertex v i ∈ V 1 , we add an edge between v i of G 1 and v π ( i ) of G 2 , this edge set is denoted by E 3 , 3 G π = ( V 1 ∪ V 2 , E 1 ∪ E 2 ∪ E 3 ). π = (123) G π = (123) G Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 3 / 1

Permutation graphs Definition Given a graph G on n vertices and a permutation π of [ n ], we define the permutation graph G π as follows : 1 we take 2 disjoint copies G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) of G , 2 for every vertex v i ∈ V 1 , we add an edge between v i of G 1 and v π ( i ) of G 2 , this edge set is denoted by E 3 , 3 G π = ( V 1 ∪ V 2 , E 1 ∪ E 2 ∪ E 3 ). G 1 G 2 π = (123) G Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 3 / 1

Permutation graphs Definition Given a graph G on n vertices and a permutation π of [ n ], we define the permutation graph G π as follows : 1 we take 2 disjoint copies G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) of G , 2 for every vertex v i ∈ V 1 , we add an edge between v i of G 1 and v π ( i ) of G 2 , this edge set is denoted by E 3 , 3 G π = ( V 1 ∪ V 2 , E 1 ∪ E 2 ∪ E 3 ). v π (1) v 1 v π (2) v 2 G 1 G 2 π = (123) v π (3) v 3 G Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 3 / 1

Permutation graphs Definition Given a graph G on n vertices and a permutation π of [ n ], we define the permutation graph G π as follows : 1 we take 2 disjoint copies G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) of G , 2 for every vertex v i ∈ V 1 , we add an edge between v i of G 1 and v π ( i ) of G 2 , this edge set is denoted by E 3 , 3 G π = ( V 1 ∪ V 2 , E 1 ∪ E 2 ∪ E 3 ). π = (123) G π G Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 3 / 1

Connectivity of permutation graphs Definition Edge-connectivity of G : λ (G) = min { d G ( X ) : ∅ � = X ⊂ V } , Minimum degree of G : δ (G) = min { d G ( v ) : v ∈ V } . Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 4 / 1

Connectivity of permutation graphs Definition Edge-connectivity of G : λ (G) = min { d G ( X ) : ∅ � = X ⊂ V } , Minimum degree of G : δ (G) = min { d G ( v ) : v ∈ V } . Remark λ ( G π ) ≤ δ ( G π ) = δ ( G ) + 1 . Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 4 / 1

Connectivity of permutation graphs Definition Edge-connectivity of G : λ (G) = min { d G ( X ) : ∅ � = X ⊂ V } , Minimum degree of G : δ (G) = min { d G ( v ) : v ∈ V } . Remark λ ( G π ) ≤ δ ( G π ) = δ ( G ) + 1 . Theorem (Goddard, Raines, Slater) For a simple graph G without isolated vertices, there exists a permutation π such that λ ( G π ) = δ ( G ) + 1 if and only if G � = 2 K k for some odd k. 2 K 3 Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 4 / 1

Necessity Theorem (Goddard, Raines, Slater) For a simple graph G without isolated vertices, there exists a permutation π such that λ ( G π ) = δ ( G ) + 1 if and only if G � = 2 K k for some odd k. 2 K 3 2 K 3 Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 5 / 1

Necessity Theorem (Goddard, Raines, Slater) For a simple graph G without isolated vertices, there exists a permutation π such that λ ( G π ) = δ ( G ) + 1 if and only if G � = 2 K k for some odd k. 2 K 3 Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 5 / 1

Necessity Theorem (Goddard, Raines, Slater) For a simple graph G without isolated vertices, there exists a permutation π such that λ ( G π ) = δ ( G ) + 1 if and only if G � = 2 K k for some odd k. 2 K 3 Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 5 / 1

Necessity Theorem (Goddard, Raines, Slater) For a simple graph G without isolated vertices, there exists a permutation π such that λ ( G π ) = δ ( G ) + 1 if and only if G � = 2 K k for some odd k. 2 K 3 Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 5 / 1

Sufficiency : Idea Theorem (Goddard, Raines, Slater) For a simple graph G without isolated vertices, there exists a permutation π such that λ ( G π ) = δ ( G ) + 1 if and only if G � = 2 K k for some odd k. G Extension : λ ( H ) = δ ( G ) + 1 , Splitting off : between G 1 and G 2 , maintaining edge-connectivity Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 6 / 1

Sufficiency : Idea Theorem (Goddard, Raines, Slater) For a simple graph G without isolated vertices, there exists a permutation π such that λ ( G π ) = δ ( G ) + 1 if and only if G � = 2 K k for some odd k. G 1 G 2 Extension : λ ( H ) = δ ( G ) + 1 , Splitting off : between G 1 and G 2 , maintaining edge-connectivity Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 6 / 1

Sufficiency : Idea Theorem (Goddard, Raines, Slater) For a simple graph G without isolated vertices, there exists a permutation π such that λ ( G π ) = δ ( G ) + 1 if and only if G � = 2 K k for some odd k. G 1 G 2 Extension : λ ( H ) = δ ( G ) + 1 , Splitting off : between G 1 and G 2 , maintaining edge-connectivity Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 6 / 1

Sufficiency : Idea Theorem (Goddard, Raines, Slater) For a simple graph G without isolated vertices, there exists a permutation π such that λ ( G π ) = δ ( G ) + 1 if and only if G � = 2 K k for some odd k. s G 1 G 2 H Extension : λ ( H ) = δ ( G ) + 1 , Splitting off : between G 1 and G 2 , maintaining edge-connectivity Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 6 / 1

Sufficiency : Idea Theorem (Goddard, Raines, Slater) For a simple graph G without isolated vertices, there exists a permutation π such that λ ( G π ) = δ ( G ) + 1 if and only if G � = 2 K k for some odd k. G 1 G 2 Extension : λ ( H ) = δ ( G ) + 1 , Splitting off : between G 1 and G 2 , maintaining edge-connectivity Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 6 / 1

Splitting off in graphs Given : graph H = ( V + s , E ), partition P = { P 1 , P 2 } of V , integer k . Definition Splitting off at s : replacing { su , sv } by uv . Complete splitting off at s : a sequence of splitting off isolating s . it is k -admissible if H ′ − s is k -edge-connected. it is P -allowed if the new edges are between P 1 and P 2 . s H Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs Given : graph H = ( V + s , E ), partition P = { P 1 , P 2 } of V , integer k . Definition Splitting off at s : replacing { su , sv } by uv . Complete splitting off at s : a sequence of splitting off isolating s . it is k -admissible if H ′ − s is k -edge-connected. it is P -allowed if the new edges are between P 1 and P 2 . s P Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs Given : graph H = ( V + s , E ), partition P = { P 1 , P 2 } of V , integer k . Definition Splitting off at s : replacing { su , sv } by uv . Complete splitting off at s : a sequence of splitting off isolating s . it is k -admissible if H ′ − s is k -edge-connected. it is P -allowed if the new edges are between P 1 and P 2 . s u v Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs Given : graph H = ( V + s , E ), partition P = { P 1 , P 2 } of V , integer k . Definition Splitting off at s : replacing { su , sv } by uv . Complete splitting off at s : a sequence of splitting off isolating s . it is k -admissible if H ′ − s is k -edge-connected. it is P -allowed if the new edges are between P 1 and P 2 . s u v Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs Given : graph H = ( V + s , E ), partition P = { P 1 , P 2 } of V , integer k . Definition Splitting off at s : replacing { su , sv } by uv . Complete splitting off at s : a sequence of splitting off isolating s . it is k -admissible if H ′ − s is k -edge-connected. it is P -allowed if the new edges are between P 1 and P 2 . s H Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 7 / 1

Splitting off in graphs Given : graph H = ( V + s , E ), partition P = { P 1 , P 2 } of V , integer k . Definition Splitting off at s : replacing { su , sv } by uv . Complete splitting off at s : a sequence of splitting off isolating s . it is k -admissible if H ′ − s is k -edge-connected. it is P -allowed if the new edges are between P 1 and P 2 . s Z. Szigeti (G-SCOP, Grenoble) Permutation hypergraphs 29 september 2010 7 / 1