Abstract
Simple finite connected graphs G=V,E of p≥2 vertices are considered in this paper. A connected detour set of G is defined as a subset S⊆V such that the induced subgraph GS is connected and every vertex of G lies on a u−v detour for some u,v∈S. The connected detour number cdnG of a graph G is the minimum order of the connected detour sets of G. In this paper, we determined cdnG for three special classes of graphs G, namely, unicyclic graphs, bicyclic graphs, and cog-graphs for Cp, Kp, and Km,n.
Highlights
For basic definitions of the concepts of graphs we refer to [1,2,3,4], and for detour distance and related terminologies in graphs, we refer to [5,6,7]
A (p, q) graph is bicyclic if and only if q p + 1. us, if G is a connected bicyclic graph, G contains either three cycles having some edges in common or contains exactly two cycles having no edges in common. e connected detour number for a block bicyclic graph is determined by the following result
Let Kp be a complete graph of order p ≥ 3, for every pair u, v of vertices, every edge other than uv lies on a u − v detour of Kp
Summary
For basic definitions of the concepts of graphs we refer to [1,2,3,4], and for detour distance and related terminologies in graphs, we refer to [5,6,7]. If S is a detour set of G and the induced subgraph G[S] is connected, S is called connected detour set (denoted c.d.s.) of G. e connected detour number of G denoted as cdn(G) is defined as cdn(G) min{|S| : S is a c.d.s. of G}. A simple connected (p, q) graph G with p ≥ 3 is called unicyclic graph iff p q. Ere are many research papers on connected detour number and edge detour graphs (see [10,11,12,13,14]). Is motivated us to determine connected detour number for other classes of graphs. Erefore, in this paper we determine the connected detour numbers for unicyclic graphs and bicyclic graphs. Journal of Mathematics the class of graphs called cog-graphs Gc will be explained and determined the cnd(Gc) if G is a complete graph, tree, cycle graph, and complete bipartite graph
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