The connected detour 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 longest x − y monophonic path is called an x − y detour monophonic path. A set S of vertices of G is a detour monophonic set of G if each vertex v of G lies on an x − y detour monophonic path, for some x and y in S. The minimum cardinality of a detour monophonic set of G is the detour monophonic number of G and is denoted by dm(G). A connected detour monophonic set of G is a detour monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected detour monophonic set of G is the connected detour monophonic number of G and is denoted by dmc(G). We determine bounds for dmc(G) and characterize graphs which realize these bounds. It is shown that for positive integers r, d and k ≥ 6 with r < d, there exists a connected graph G with monophonic radius r, monophonic diameter d and dmc(G) = k. For each triple a, b, p of integers with 3 ≤ a ≤ b ≤ p − 2, there is a connected graph G of order p, dm(G) = a and dmc(G) = b. Also, for every pair a, b of positive integers with 3 ≤ a ≤ b, there is a connected graph G with mc(G) = a and dmc(G) = b, where mc(G) is the connected monophonic number of G.