Edge-connectivity and super edge-connectivity of P2-path graphs

Camino Balbuena, Daniela FerreroDiscrete Mathematics 269 (2003) 13 – 20

Outline

Introduction P2-path graph

Result 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.

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.

1 4

2 3

<1,2,3> <2,3,4>

<4,1,2>

<3,4,1>

Example:

<2,3,1><3,1,4>

<1,3,4>

<2,1,3>

P2-path graph

Path graphs were investigated by Broersma and Hoede [6] as a natural generalization of line graphs.

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.

A

B

1

2

3 45

6

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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)

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

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

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-

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.

e

f

g

h

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 g

enerated 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 t

o 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 som

e 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)

a

bc

d

e

f

g

y ts x

a

b

c

d

e

f

g

ytsx

Lemma

Deletion of an edge reduces connectivity by at most 1