The Connected Monophonic Number of a Graph

Abstract

For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A set S of vertices of G is a monophonic set of G if each vertex v of G lies on an 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 connected monophonic set of G is a monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected monophonic set of G is the connected monophonic number of G and is denoted by mc(G). We determine bounds for it and characterize graphs which realize these bounds. For any two vertices u and v in G, the monophonic distance dm(u, v) from u to v is defined as the length of a longest u − v monophonic path in G. The monophonic eccentricity em(v) of a vertex v in G is the maximum monophonic distance from v to a vertex of G. The monophonic radius radmG of G is the minimum monophonic eccentricity among the vertices of G, while the monophonic diameter diammG of G is the maximum monophonic eccentricity among the vertices of G. It is shown that for positive integers r, d and n ≥ 5 with r < d, there exists a connected graph G with radmG = r, diammG = d and mc(G) = n. Also, if a,b and p are positive integers such that 2 ≤ a < b ≤ p, then there exists a connected graph G of order p, m(G) = a and mc(G) = b.