An inequality on global alliances for trees

TL;DR

This paper establishes a new inequality relating the sizes of minimum global defensive and offensive alliances in trees, addressing an open question in the field of graph theory.

Contribution

It proves an inequality connecting the minimum sizes of global defensive and offensive alliances in trees, providing a theoretical advancement in alliance theory.

Findings

Proves an inequality between alliance sizes in trees

Answers an open question in alliance theory

Advances understanding of dominating sets in graphs

Abstract

In this paper, we prove an inequality on the cardinalities of the minimum size global defensive alliance and the minimum size global offensive alliance. A global defensive alliance is a dominating set such that when any point inside a selected group $S$ is chosen, at least half of the points in its neighborhood are also in the set $S$, including the selected point. A global offensive alliance is a dominating set such that if any point outside $S$ is selected, at least half of the points in its neighborhood, including the selected point, are in set $S$. Our result answers an open question in [HA].