The forcing vertex detour monophonic number of a graph
P. Titus, P. Balakrishnan · AKCE International Journal of Graphs and Combinatorics · 2016
For any two vertices and in a connected graph , an – path is a monophonic path if it contains no chord, and a longest – monophonic path is called an – detour monophonic path. For any vertex in , a set is an -detour monophonic set of if each vertex lies on an – detour monophonic path for some element in . The minimum cardinality of an -detour monophonic set of is the -detour monophonic number of , denoted by . A subset of a minimum -detour monophonic set of is an -forcing subset for if is the unique minimum -detour monophonic set containing . An -forcing subset for of minimum cardinality is a minimum -forcing subset of . The forcing -detour monophonic number of , denoted by , is the cardinality of a minimum -forcing subset for . The forcing -detour number of is , where the minimum is taken over all minimum -detour monophonic sets in . We determine bounds for it and find the same for some special classes of graphs. Also we show that for every pair of integers with , there exists a connected graph such that and for some vertex in .