Expanded-clique graphs and the domination problem

TL;DR

This paper introduces expanded-clique graphs derived from a base graph with vertex values, provides characterizations including a linear-time recognition algorithm, and explores the NP-completeness of the domination problem in certain cases.

Contribution

It offers two characterizations of expanded-clique graphs, including a linear-time recognition algorithm, and analyzes the computational complexity of the domination problem.

Findings

Characterizations of expanded-clique graphs

Linear-time recognition algorithm for these graphs

NP-completeness of the domination problem in specific graph classes

Abstract

Given a graph $G$ such that each vertex $v_i$ has a value $f(v_i)$, the expanded-clique graph $H$ is the graph where each vertex $v_i$ of $G$ becomes a clique $V_i$ of size $f(v_i)$ and for each edge $v_iv_j \in E(G)$, there is a vertex of $V_i$ adjacent to an exclusive vertex of $V_j$. In this work, among the results, we present two characterizations of the expanded-clique graphs, one of them leads to a linear-time recognition algorithm. Regarding the domination number, we show that this problem is \NP-complete for planar bipartite $3$-expanded-clique graphs and for cubic line graphs of bipartite graphs.