Boundary vertices in graphs

Discrete Mathematics 263 (2003) 25-34

Gary Chartrand, David Erwin

Garry L. Johns, Ping Zhang

Abstract

Introduce Peripheral vertex (Per) Eccentric vertex (Ecc) Boundary vertex ( )

Show there relation Per(G) Ecc(G) (G)

Outline

Introduction Boundary graphs The boundary, eccentric subgraph, and perip

hery of a graph

Eccentric vertex

Eccentricity e(v) The distance between v and a vertex farthest fro

m v

v

e(v)=3

1

1

1 2

2

2 3

3

3

# distance d(u,v) the length of a shortest u-v path

Eccentric vertex

Eccentric vertex A vertex u of G is an eccentric vertex of a vertex v

if d(u,v)=e(v) Eccentric subgraph Ecc(G)

The subgraph of G induced by its eccentric vertices

Eccentric subgraph - Ecc(G)

Ecc(G)

Peripheral vertex

Diameter The maximum eccentricity

Peripheral vertex A vertex v in a connected graph G is a peripheral

vertex if e(v) = diameter

diam(G)=4

Periphery per(G)

Periphery per(G) The subgraph of G induced by its peripheral vertic

es is the periphery per(G)

Per(G)

Boundary vertex

Boundary vertex A vertex u of G is a boundary vertex of a vertex v i

f d(w,v) d(u,v) w N(u)

1

2 1

1

1

2

2

2

3

v

u u

u

uv

w

Boundary vertex

A vertex u is a boundary vertex of G if u is the boundary vertex of some vertex of G

Boundary The subgraph of G induced by its boundary vertic

es is the boundary (G)

Boundary

(G)

Proposition 1.1

Proposition 1.1 No cut-vertex of a graph is a boundary vertex

Pf. Assume, to the contrary, that there exists a graph G and a cut-vertex u of G such that u is a boundary vertex of some vertex v in G. Let G1 and G2 be two distinct components of G-u such that v V(G1),and let w be a neighbor of u that belongs to G2. Then d(w,v)=d(u,v)+1,contrary to the hypothesis.

Def : cut-vertex of a graph is a vertex whose deletion increases the number of components

d(w,v)<=d(u,v)

Proposition 1.2

Proposition 1.2 Let v be a vertex in a connected graph G such

that v belongs to a block B and v is not a cut-vertex of G. Then v is a boundary vertex of G if and only if v is a boundary vertex of B

Def : A block of a graph is a maximal connected subgraph has no cut-vertex

Proposition 1.2

Pf: Certainly, every boundary vertex of a block of so

me connected graph is a boundary vertex of the graph. It remains then only to verify the converse. Let G be a connected graph and let v be a boundary vertex of G. Thus v is a boundary vertex of some vertex w in G. Since v is not a cut-vertex, v belongs to a unique block B of G. If w V(B),then the proof is complete. Thus; we may assume that w V(B).

Let w belong to the block B’, where then B’<>B.For each y v(B),every w-y geodesic contains a unique cut-vertex x of G that belongs to B. Hence d(w,v)=d(w,x)+d(x,v).Let u N(v).Then u V(B) and so d(w,u)=d(w,x)+d(x,u).Because v is a boundary vertex of w, it follows that d(w,u)<=d(w,v). Therefore, d(x,u)<=d(x,v),which implies that v is a boundary vertex of x as well

Proposition 1.3

Proposition 1.3 Let G be a connected graph. A vertex v of G is a

boundary vertex of every vertex distinct from v if and only if v is a complete vertex of G.

Def : A vertex in a graph is called complete if the subgraph induced by its neighborhood is complete

Proposition 1.3

Pf: First, Let v be a complete vertex in G and let w be

a vertex distinct from v.Let w=v0,v1,v2,…,vk=v be a w-v geodesic. Let u be a neighbor of v. If u=vk-1, then d(w,u)<d(w,v).So we may assume that u<>vk-1. Since v is complete,uvk-1 E(G) and w=v0,v1,v2,…,vk-1,u is a w-u path in G, implying that d(w,u)<=d(w,v).Hence v is a boundary vertex of w.

For the converse, let v be a vertex of G that is not a complete vertex. Then there exist distinct, nonadjacent vertices u,w N(v), Since d(u,w)>d(u,v),it follows that v is not a boundary vertex of u.

Proposition 1.4

Proposition 1.4 Let G be nontrivial connected graph and let u be a

vertex of G. Every vertex distinct from u is a boundary vertex of u if and only if e(u)=1

Pf: Assume first that e(u)=1 and let v be a vertex of G

distinct from u. Let w be a neighbor of v, Then d(u,w)<=1 and d(u,v)=1. Hence v is a boundary vertex of u. For the converse, assume, to the contrary, that every vertex of G different from u is a boundary vertex of u but e(u)<>1.Then there exists a vertex x in G such that d(x,u)=2. Let x,y,u be a path in G. Then u is a neighbor of y and d(x,u)=2, while d(y,u)=1. Thus y is not a boundary vertex of u , which is a contradiction.

Boundary graphs

Per(G) Ecc(G) (G) G means subgraph

characterization

Theorem A. A nontrivial graph F is the periphery of some

connected graph if and only if every vertex of F has eccentricity 1 or no vertex of F has eccentricity 1

Theorem B. A nontrivial graph F is the eccentric of some

connected graph if and only if every vertex of F has eccentricity 1 or no vertex of F has eccentricity 1

Lemma 2.1 Let G be connected graph of diameter 2. Then ev

ery vertex v is a boundary vertex of G unless v is the unique vertex of G having eccentricity 1

Lemma 2.2 Let F be a nontrivial connected graph with no verti

ces of eccentricity 1 and let G = F + Kk where k>=1. Then G is a self-boundary graph if and only if K >=2 (?)

Theorem 2.3

Theorem 2.3 A nontrivial graph H is the boundary of some

connected graph if and only if H does not have exactly one vertex with eccentricity 1

Theorem 3.2

For each triple a,b,c of integers with 2 a b c,there is a connected graph G such that Per(G) has order a, Ecc(G) has order b, and (G) has order c.

Theorem 3.3

For each triple r,s,t of rational numbers with 0<r s t 1,there is a connected graph G of order n such that

tn

Gs

n

Gr

n

G

)(,

)(,

)(