Expanded-clique graphs and the domination problem
Mitre C. Dourado, Rodolfo A. Oliveira, Vitor Ponciano, Alessandra B. Queiróz, Rômulo L. O. Silva
TL;DR
The paper introduces expanded-clique graphs H built from a root G and a function f, where each vertex induces a clique of size f(v) and adjacent roots create cross edges; it develops two characterizations and a linear-time recognition algorithm, and studies the domination problem on this graph class. It situates expanded-clique graphs between subdivided-line graphs and line graphs of bipartite graphs, showing that Hasunuma subdivided-line graphs and Sierpiński graphs are included, and providing a forbidden-substructure (bad chain, butterfly, claw, C4, diamond, odd-hole)-free characterization. The domination results establish NP-completeness for planar bipartite $3$-expanded-clique graphs and for cubic line graphs of bipartite graphs, and reveal a tight relationship gamma(H) + alpha_2(G,f) = |V(G)| with practical approximation bounds. Collectively, the work advances both structural understanding and complexity boundaries for recognition and domination in expanded-clique graphs, with implications for related 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.