On the detection of local and global amoebas: theoretical insights and practical algorithms

Let G be a graph of order n, and let e is an element of E(G) and e' is an element of ((V(G))(2)) \ E(G). If the graph G' = G - e + e' is isomorphic to G, we say that e -> e' is a feasible edge-replacement. We call G a local amoeba if, for any two copies G(1) and G(2) of G on V(G), G(2) can be reach from G(1) by performing a sequence of edge replacements. A graph G is a global amoeba if it can be made into a local amoeba by adding some isolated vertices. Our work includes a proof that almost every graph is not an amoeba, and the identification of a special type of edge-replacement called weird-edge-replacements. Additionally, we provide an infinite family of trees that are both weird local and global amoebas. Our contributions extend to the development and implementation of several algorithms for detecting local and global amoebas, which are made available in a public repository along with multiple examples. (C) 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0)