Total outer-independent coalition in graphs

Abstract

A set $D$ of vertices graph $G$ is a total outer-independent dominating set (TOIDS) of $G$ if every vertex of $G$ is adjacent to at least one vertex in $D$, and $V(G)\setminus D$ is an independent set of $G$. A TOI-coalition in $G$ comprises two disjoint sets of vertices $A$ and $B$ of $G$, neither of which is a TOIDS but whose union $A\cup B$ is a TOIDS of $G$. We say that the sets $A$ and $B$ form a TOI-coalition, and are TOI-coalition partners. A TOI-coalition partition in $G$ is a vertex partition $\hat{I}¨=\{V_1,V_2,...,V_k\}$ in which every set forms a TOI-coalition with another set in $\hat{I}¨$. The TOI-coalition number $C_t^{oi}(G)$ is the maximum cardinality among all TOI-coalition partitions of $G$. In this work, the above-mentioned concepts are introduced and studied. The existence of a TOI-coalition partition is investigated. Several sharp upper bounds on $C_t^{oi}(G)$ are established. Finally, the exact values of $C_t^{oi}(G)$ for some graph classes are obtained.