On the monophonic global domination number of a graph

Abstract

UDC 519.17

We introduce the monophonic global domination number $\overline{\gamma}_m(G)$ by combining monophonic convexity (via chordless paths) with the global domination in a graph and its complement. A monophonic global dominating set is defined, and $\overline{\gamma}_m(G)$ is regarded as the minimum size of sets of this kind. We also establish the bounds, relate $\overline{\gamma}_m(G)$ to the classical domination number, and characterize the graphs that attain extreme values. The realization theorem is proved for prescribed parameter values. The behavior of $\overline{\gamma}_m(G)$ under graph operations, in particular, for the corona product, is analyzed. The applications to the network monitoring are discussed and several open problems are proposed for further research.