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## Proyecciones (Antofagasta)

*Print version* ISSN 0716-0917

#### Abstract

SANTHAKUMARAN, A. P. and MAHENDRAN, M.. **The forcing open monophonic number of a graph**.* Proyecciones (Antofagasta)* [online]. 2016, vol.35, n.1, pp.67-83.
ISSN 0716-0917. http://dx.doi.org/10.4067/S0716-09172016000100005.

For a connected graph G of order n ≥ 2, and for any mínimum open monophonic set S of G, a subset T of S is called a forcing subset for S if S is the unique minimum open monophonic set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing open monophonic number of S, de-noted by f_{om}(S), is the cardinality of a minimum forcing subset of S. The forcing open monophonic number of G, denoted by f_{om}(G), is f_{om}(G) = min(f_{om}(S)), where the minimum is taken over all minimum open monophonic sets in G. The forcing open monophonic numbers of certain standard graphs are determined. It is proved that for every pair a, b of integers with 0 ≤ a ≤ b - 4 and b ≥ 5, there exists a connected graph G such that f_{om}(G) = a and om(G) = b. It is analyzed how the addition of a pendant edge to certain standard graphs affects the forcing open monophonic number.

**Keywords
:
**Monophonic number; open monophonic number; forcing monophonic number; forcing open monophonic number.