Total restrained coalitions in graphs
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 se...
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Zusammenfassung: | 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 $\Phi=\{V_1, V_2,\dots,V_k\}$ 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. |
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DOI: | 10.48550/arxiv.2412.18623 |