Enumeration of Glued Graphs of Paths - Ruangnai, Monthiya; Panma, Sayan (Communications in Mathematics and Applications, 2019)
Let \(G_1\) and \(G_2\) be two vertex-disjoint graphs with \(H_1\) a subgraph of \(G_1\) and \(H_2\) a subgraph of \(G_2\). Let \(f:H_1 → H_2\) be an isomorphism between these subgraphs. The glued graph of \(G_1\) and \(G_2\) at \(H_1\) and \(H_2\) with respect to \(f\) is the graph that results from combining \(G_1 \cup G_2\) by identifying the subgraphs \(H_1\) and \(H_2\) according to the isomorphism \(f\) between \(H_1\) and \(H_2\). We refer \(G_1\) and \(G_2\) as its original graphs and refer \(H\) as its clone where \(H\) is a copy of \(H_1\) and \(H_2\). In this paper, we enumerate all non-isomorphic resulting glued graphs between two paths at connected clones. Moreover, we also give the characterization of the glued graph at a connected clone.
Keywords: A glued graph; The glue operator; Glued graph of paths; Graphs enumeration; Graph isomorphisms