# Forcing Total Detour Monophonic Sets in a Graph

## Abstract

For a connected graph G = (V,E) of order at least two, a totaldetour monophonic set of a graph G is a detour monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dmt(G). A subset T of a minimum total detour monophonic set S of G is a forcing total detour monophonic subset for S if S is

the unique minimum total detour monophonic set containing T. A forcing total detour monophonic subset for S of minimum cardinality is a minimum forcing total detour monophonic subset of S. The forcing total detour monophonic number ftdm(S) in G is the cardinality of a minimum forcing total detour monophonic subset of S. The forcing total detour monophonic number of G

is ftdm(G) = min{ftdm(S)}, where the minimum is taken over all minimum total detour monophonic sets S in G. We determine bounds for it and find the forcing total detour monophonic number of certain classes of graphs. It is

shown that for every pair a, b of positive integers with 0 6 a < b and b > 2a+1, there exists a connected graph G such that ftdm(G) = a and dmt(G) = b.

## References

[1] F. Buckley and F. Harary. Distance in Graphs, Addison-Wesley, Redwood City, CA, 1990.

[2] F. Harary. Graph Theory. Addison-Wesley, 1969.

[3] P. Titus and K. Ganesamoorthy. On the Detour Monophonic Number of a Graph. Ars Combinatoria, 129(2016), 33-42.

[4] P. Titus, K. Ganesamoorthy and P. Balakrishnan. The Detour Monophonic Number of a Graph. J. Combin. Math. Combin. Comput., 84(2013), 179-188.

[5] A.P. Santhakumaran, P. Titus, and K. Ganesamoorthy. The Total Detour Monophonic Number of a Graph, Proyecciones Journal of Mathematics, (to appear).

[2] F. Harary. Graph Theory. Addison-Wesley, 1969.

[3] P. Titus and K. Ganesamoorthy. On the Detour Monophonic Number of a Graph. Ars Combinatoria, 129(2016), 33-42.

[4] P. Titus, K. Ganesamoorthy and P. Balakrishnan. The Detour Monophonic Number of a Graph. J. Combin. Math. Combin. Comput., 84(2013), 179-188.

[5] A.P. Santhakumaran, P. Titus, and K. Ganesamoorthy. The Total Detour Monophonic Number of a Graph, Proyecciones Journal of Mathematics, (to appear).

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## Published

2017-04-28

## Issue

## Section

Чланци