Edge-connectivity and super edge-connectivity of P 2 -path graphs Camino Balbuena, Daniela Ferrero Discrete Mathematics 269 (2003) 13 – 20

Outline  Introduction Introduction  P 2 -path graph P 2 -path graph  Result Result  Review:Line Graph Review:Line Graph

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.

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.

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.

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.

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).

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.

P 2 -path graph  Given a graph G, the vertex set of the P 2 (G)-path graph is the set of all paths of length two of G.  Two vertices of P 2 (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.

14 23 Example:

P 2 -path graph  Path graphs were investigated by Broersma and Hoede [6] as a natural generalization of line graphs.

P 2 -path graph Theorem A:(M. Knor, 2001) Let G be a connected graph. Then P 2 (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.

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

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

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

Result (Theorem 2.3) Let G be a -regular graph with (G)  4. Then P 2 (G) is optimum super- and 1 (P 2 (G)) = 4 − 6. Note: about optimum super-

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.

 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) Not a line graph

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.

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)

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.

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.

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.

Ford-Fulkerson, 1956   ’ G (x,y)= L(G’) (sx,yt)= L(G’) (sx,yt)= ’ G (x,y)

local equality  ’ (x,y): the maximum size of a set of pairwise edge-disjoint x,y-paths   ’ (x,y): the minimum number of edges whose deletion makes y unreachable from x.  Ford-Fulkerson proved that always ’ (x,y)=  ’ (x,y)

Lemma  Deletion of an edge reduces connectivity by at most 1

Theorem  The connectivity of G equals the maximum  such that (x,y) for all x,y V(G).  The edge-connectivity of G equals the maximum  such that ’ (x,y) for all x,y V(G).