On \(K_*\)-Ultrahomogeneous Graphs

Abstract

Let \(\mathcal{C}\) be any class of finite graphs. A graph \(G\) is \(\mathcal{C}\)-ultrahomogeneous if every isomorphism between induced subgraphs belonging to \(\mathcal{C}\) extends to an automorphism of \(G\). We study finite graphs that are \({K}_*\)-ultrahomogeneous, where \({K}_*\) is the class of complete graphs. We also explicitly classify the finite graphs that are \(\sqcup{K}_{*}\)-ultrahomogeneous, where \(\sqcup{K}_{*}\) is the class of disjoint unions of complete graphs.