Total Offensive Alliances on Some Graphs

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 o�ensive 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.