The Connected Open Monophonic Number of a Graph
A. P. Santhakumaran, M. Mahendran · International Journal of Computer Applications · 2013
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 asis 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, om c (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 om c (G) = k.