Secure coalitions in graphs
Swathi Shetty, Sayinath Udupa N. V., B. R. Rakshith
TL;DR
The paper develops the theory of secure coalitions in graphs by introducing sec-partitions and the associated secure coalition graph $SCG(G,\pi)$. It establishes existence results, derives bounds relating $SEC(G)$ to the coalition number $C(G)$, and provides sharp lower bounds tied to the graph's minimum degree. It fully characterizes graphs with $SEC(G)=n$, distinguishing connected cases via four families and showing disjoint unions of complete graphs for the disconnected case, while also detailing the tree situation, where only small paths reach $SEC(T)=n$ and $SEC(T)=n-1$ occurs for specific trees. Finally, it proves every graph without isolated vertices is a secure coalition graph through a constructive embedding, highlighting the rich interplay between secure domination, partitioning, and coalition graphs with potential implications for graph partitioning problems and network security models.
Abstract
A secure coalition in a graph $G$ consists of two disjoint vertex sets $V_1$ and $V_2$, neither of which is a secure dominating set, but whose union $V_1 \cup V_2$ forms a secure dominating set. A secure coalition partition ($sec$-partition) of $G$ is a vertex partition $π= \{V_1, V_2, \dots, V_k\}$ where each set $V_i$ is either a secure dominating set consisting of a single vertex of degree $n-1$, or a set that is not a secure dominating set but forms a secure coalition with some other set $V_j \in π$. The maximum cardinality of a secure coalition partition of $G$ is called the secure coalition number of $G$, denoted $SEC(G)$. For every $sec$-partition $π$ of a graph $G$, we associate a graph called the secure coalition graph of $G$ with respect to $π$, denoted $SCG(G,π)$, where the vertices of $SCG(G,π)$ correspond to the sets $V_1, V_2, \dots, V_k$ of $π$, and two vertices are adjacent in $SCG(G,π)$ if and only if their corresponding sets in $π$ form a secure coalition in $G$. In this study, we prove that every graph admits a $sec$-partition. Further, we characterize the graphs $G$ with $SEC(G) \in \{1,2,n\}$ and all trees $T$ with $SEC(T) = n-1$. Finally, we show that every graph $G$ without isolated vertices is a secure coalition graph.