On the toll number of a graph

Abstract

A tolled walk T between two different, non-adjacent vertices u , v in a graph G is a walk in which u and v have exactly one neighbor. A toll interval between u , v ∈ V ( G ) , T ( u , v ) , is the set of all vertices of G that lie in some tolled u , v -walk. A set S ⊆ V ( G ) is t -convex if T ( u , v ) ⊆ S for any u , v ∈ S . The size of a maximum proper t -convex set S ⊆ V ( G ) is called the t -convexity number of a graph G and is denoted by c t ( G ) . A vertex v ∈ V ( G ) is t -extreme if V ( G ) − { v } is a t -convex set of G . The toll number of G , t n ( G ) , is the smallest size of a set S with ⋃ u , v ∈ S T ( u , v ) = V ( G ) . In this paper we prove that t n ( G ) ≤ 6 in any prime graph G with diam ( G ) > 2 . Then we present the exact value of the toll number of a cograph which implies that in any cograph G , t n ( G ) can be computed in polynomial time. Cographs are used to present the existence of a graph G of diameter 2 in which the difference between t n ( G ) and the number of t -extreme vertices is arbitrary large. Then we consider graphs with extreme toll number and give a characterization of graphs G with t n ( G ) ∈ { 2 , | V ( G ) | − 1 , | V ( G ) | } . Finally we give a characterization of t -convex sets and prove that for a given k , deciding whether c t ( G ) ≥ k is NP-complete.