On $K_{\ast }$-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 $K_{\ast }$-ultrahomogeneous, where $K_{\ast }$ is the class of complete graphs. We also explicitly classify the finite graphs that are $\bigsqcup K_{\ast }$-ultrahomogeneous, where $\bigsqcup K_{\ast }$ is the class of disjoint unions of complete graphs.