Total and forcing total edge-to-vertex monophonic number of a graph

摘要

For a connected graph , a set is called a total edge-to-vertex monophonic set of a connected graph G if the subgraph induced by S has no isolated edges. The total edge-to-vertex monophonic number of G is the minimum cardinality of its total edge-to-vertex monophonic set of G. The total edge-to-vertex monophonic number of certain classes of graphs is determined and some of its general properties are studied. Connected graphs of size with total edge-to-vertex monophonic number q is characterized. It is shown that for positive integers and with , there exists a connected graph G with , and and also shown that for every integers a and b with , there exists a connected graph G such that and . A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing total edge-to-vertex monophonic number of S, denoted by is the cardinality of a minimum forcing subset of S. The forcing total edge-to-vertex monophonic number of G, denoted by , where the minimum is taken over all total edge-to-vertex monophonic set S in G. The forcing total edge-to-vertex monophonic number of certain classes of graphs are determined and some of its general properties are studied. It is shown that for every integers a and b with and , there exists a connected graph G such that and , where is the forcing total edge-to-vertex monophonic number of G.