Total restrained coalitions in graphs
M. Chellali, J. C. Valenzuela-Tripodoro, H. Golmohammadi, I. I. Takhonov, N. A. Matrokhin
TL;DR
We introduce the notion of total restrained coalitions in graphs, formalizing TRD-sets and trc-partitions, and define the total restrained coalition number $C_{tr}(G)$ as the maximum size of a trc-partition. We establish core properties and bounds, show $C_{tr}(G)=0$ if $G$ has an isolated vertex and that isolate-free graphs admit a trc-partition with $C_{tr}(G)\ge 2d_t^r(G)$, with $d_t^r(G)=d_t(G)$. We compute exact values for several graph families (e.g., $C_{tr}(K_n)=n$, $C_{tr}(K_{p,q})=n$, $C_{tr}(K_{1,n-1})=2$, and explicit results for $P_n$ and $C_n$) and present structural characterizations for graphs with maximal $C_{tr}(G)$. The results yield insights into large-$C_{tr}$ graphs, including a universal-vertex condition, diameter constraints, and complete triangle-free and tree classifications, with numerous directions for future work.
Abstract
A set $S\subseteq V$ in an isolate-free graph $G$ is a total restrained dominating set, abbreviated TRD-set, if every vertex in $V$ is adjacent to a vertex in $S$, and every vertex in $V\setminus S$ is adjacent to a vertex in $V\setminus S$. A total restrained coalition is made up of two disjoint sets of vertices $X$ and $Y$ of $G$, neither of which is a TRD-set but their union $X\cup Y$ is a TRD-set. A total restrained coalition partition of a graph $G$ is a partition $Φ=\{V_1, V_2,\dots,V_k\}$ such that for all $i \in [k]$, the set $V_i$ forms a total restrained coalition with another set $V_j$ for some $j$, where $j\in [k]\setminus{i}$. The total restrained coalition number $C_{tr}(G)$ in $G$ equals the maximum order of a total restrained coalition partition in $G$. In this work, we initiate the study of total restrained coalition in graphs and its properties.