Total Offensive Alliances on Some Graphs
Maxene S. Hablo *
Mathematics Department, College of Arts and Sciences, Central Mindanao University, Musuan, Maramag, Bukidnon, Philippines.
Isagani S. Cabahug Jr.
Mathematics Department, College of Arts and Sciences, Central Mindanao University, Musuan, Maramag, Bukidnon, Philippines.
*Author to whom correspondence should be addressed.
Abstract
Let G = (V (G), E(G)) be a nontrivial connected graph. A nonempty set of vertices T \(\subseteq\) V (G) is deffned as an offensive alliance in G if, for every v \(\in\) \(\partial\)(T), it holds that \(\mid N[v] \cap\) T| \(\ge\) | N[v] \ T|. Equivalently, this can be expressed as degT \((v) \ge\) degv(G)\T (v) + 1. The set T is termed a total offensive alliance in G if it is an offensive alliance and every vertex in T has at least one neighbor within T. The minimum cardinality of a total oensive alliance set in G is called the total offensive alliance number, denoted by ato(G). This paper presents a characterization of total offensive alliance sets and provides the corresponding minimum cardinality for various graph families, including path, cycle, complete, star, fan, and wheel graphs.
Keywords: Offensive alliance, total offensive alliance, total offensive alliance number