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.

翻译：给定一个图$G$，其中每个顶点$v_i$具有值$f(v_i)$，扩展团图$H$定义为：$G$中每个顶点$v_i$变为大小为$f(v_i)$的团$V_i$，且对于$G$中的每条边$v_iv_j \in E(G)$，存在$V_i$中的一个顶点与$V_j$中的一个独占顶点相邻。在本研究中，我们提出了扩展团图的两种刻画，其中一种可导出线性时间识别算法。关于支配数问题，我们证明该问题对于平面二分$3$-扩展团图以及二分图的立方线图是\NP完全的。