Edge-connectivity and super edge-connectivity of P 2 -path graphs

1. Edge-connectivity and super edge-connectivity of P2-path graphs Camino Balbuena, Daniela Ferrero Discrete Mathematics 269 (2003) 13 – 20

2. Outline • Introduction • P2-path graph • Result • Review:Line Graph

3. Introduction • A graph G is called connected if every pair of vertices is joined by a path. • An edge cut in a graph G is a set T of edges of G such that G − T is not connected.

4. Introduction • If T is a minimal edge cut of a connected graph G, then, G − T necessarily contains exactly two components. • It is usual to denote an edge cut T as (C, Ĉ), where C is a proper subset of V(G) and (C, Ĉ) denotes the set of edges between C and its complement Ĉ.

5. Introduction • A minimum edge cut (C, Ĉ) is called trivial if C = {v} or Ĉ = {v} for some vertex v of deg(v) = (G). • The edge-connectivity, (G), of a graph G is the minimum cardinality of an edge cut of G.

6. Introduction • A graph G is said to be maximally edge-connected when (G) = (G). • A maximally edge-connected graph is called super-  if every edge cut (C, Ĉ) of cardinality (G) satisfes that either |C|=1 or |Ĉ|=1.

7. Introduction • 1(G) = min{|(C, Ĉ)|, (C, Ĉ) is a nontrivial edge cut}. (conditional) • A graph G is super- if and only if 1(G)> (G). • The edge-superconnectivity of a graph G is the value of 1(G).

8. Introduction • Furthermore, 1(G) min{deg(u) + deg(v), e=uv∈E(G)}−2=M. • G is said to be optimum super-, if every minimum nontrivial edge cut is the set of edges incident with some edge of G. In this case, 1(G) = M 2(G) − 2.

9. P2-path graph • Given a graph G, the vertex set of the P2(G)-path graph is the set of all paths of length two of G. • Two vertices of P2(G) are joined by an edge, if and only if, the intersection of the corresponding paths form an edge of G, and their union forms either a cycle or a path of length 3.

10. Example: <1,2,3> <2,3,4> 1 4 <2,1,3> <3,4,1> 2 3 <1,3,4> <4,1,2> <3,1,4> <2,3,1>

11. P2-path graph • Path graphs were investigated by Broersma and Hoede [6] as a natural generalization of line graphs.

12. P2-path graph Theorem A:(M. Knor, 2001) Let G be a connected graph. Then P2(G) is disconnected if and only if G contains two distinct paths A and B of length two, such that the degrees of both end vertices of A are 1 in G.

13. By Theorem A • If G is a connected graph with at most one vertex of degree one, then P2(G) is also connected. • Result 1:Theorem 2.1 ((G)2, (G)  2)

14. Result (Theorem 2.1) Let G be a connected graph with (G)2. Then, (a) (P2(G)) (G) − 1, (b) (P2(G))  2 (G) − 2 if (G)  2. Note: (P2(G))=2(G)−2 for regular graphs (P2(G))2(G)−2 in general Best possible at least for regular graphs

15. Result (Theorem 2.2) Let G be a graph with (G)  3, such that (P2(G))=2(G)−2. Then P2(G) is super- and 1(P2(G))  3((G) − 1). Note: about superconnectivity

16. Result (Theorem 2.3) Let G be a -regular graph with (G)  4. Then P2(G) is optimum super- and 1(P2(G)) = 4 − 6. Note: about optimum super- 

17. Line graph (Definition) The line graph of a graph G, written L(G), is the graph whose vertices are the edges of G, with ef E(L(G)) when e=uv and f=vw in G. G L(G)

18. If G=L(H) and H is simple, then each v V(H) with d(v)2 generates a clique Q(v) in G corresponding to edges incident to v. • These cliques partition E(G). • Each vertex e V(G) belongs only to the cliques generated by the two endpoints of e E(H) H G=L(H) Not a line graph

19. Property 1 (Krausz, 1943) • For a simple graph G, there is a solution to L(H)=G if and only if G decomposes into complete subgraphs, with each vertex of G appearing in at most two in the list.

20. Property 2 (van Rooij and Wilf, 1965) • For a simple graph G, there is a solution to L(H)=G if and only if G is claw-free and no double triangle of G has two odd triangles. • An induced kite is a double triangle; it consists of two triangles sharing an edge, and the two vertices not in that edge are nonadjacent. • T is odd if |N(v) V(T)| is odd for some v V(G) • T is even if |N(v) V(T)| is even for every v V(G)

21. Property 3 (Beineke, 1968) • A simple graph G is the line graph of some simple graph if and only if G does not have any of the nine graphs below as an induced subgraph.

22. local equality • Menger stated the local equality (x,y)= (x,y) • (x,y): the minimum size of an x,y-cut. • (x,y): the maximum size of a set of pairwise internally disjoint x,y-paths.

23. Theorem (’(x,y)= ’(x,y)) • If x and y are distinct vertices of a graph G, then the minimum size of an x, y-disconnecting set of edges equals the maximum number of pairwise edge-disjoint x, y-paths.

24. Ford-Fulkerson, 1956 • ’G(x,y)=L(G’)(sx,yt)=L(G’)(sx,yt)=’G(x,y) L(G') G G'

25. Lemma • Deletion of an edge reduces connectivity by at most 1