The Connected Open Monophonic Number of a Graph

Abstract

In this paper, we introduce and investigate the connected open monophonic sets and related parameters. For a connected graph G of order n, a subset S of vertices of G is a monophonic set of G if each vertex v in G lies on a x-y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G). A monophonic set of cardinality m(G) is called a m–set of G. A set S of vertices of a connected graph G is an open monophonic set of G if for each vertex v in G, either v is an extreme vertex of G and v  S, or v is an internal vertex of a x-y monophonic path for some x, y  S. An open monophonic set of minimum cardinality is a minimum open monophonic set and this cardinality is the open monophonic number, om(G). A connected open monophonic set of G is an open monophonic set S such that the subgraph induced by S is connected. The minimum cardinality of a connected open monophonic set of G is the connected open monophonic number, omc(G). Certain general properties satisfied by connected open monophonic sets are investigated. The connected open monophonic numbers of certain standard graphs are determined. A necessary condition for the connected open monophonic number of a graph G of order n to be n is determined. A graph with connected open monophonic number 2 is characterized. It is proved that for any k, n of integers with 3 ≤ k ≤ n, there exists a connected graph G of order n such that omc(G) = k.