Stem-Symmetry, Comb Products, and their Relation to Amoeba Graphs

TL;DR

This work develops a group-theoretic framework for labeled-graph amoebas, introducing the Fer group to encode feasible edge-replacements and the hang group to capture how local amoebas embed into larger ones. It establishes equivalences between hang-symmetry and stem-symmetry and provides a transitivity-based criterion for global amoebas, linking these symmetries to structural graph operations such as leaf/isolated-vertex augmentation. The comb product is analyzed through wreath-product embeddings, yielding constructive conditions under which comb-products preserve local or global amoeba properties, and revealing recursive pathways to generate broad families of amoebas. The paper also refines and extends previous results, showing preservation and limitations of stem-/hang-symmetry under comb products, and highlights open questions about primitive-like Fer-group realizations and broader graph-operator preservations. Overall, the results broaden the toolkit for constructing and classifying local/global amoebas via permutation-group actions and hierarchical graph operations.

Abstract

Local and global amoebas are families of labeled graphs that satisfy interpolation properties on a fixed vertex set. A labeled graph $G$ on $n$ vertices is a local amoeba (resp. global amoeba) if there exists a sequence of feasible edge-replacements between any two labelled embeddings of $G$ into $K_n$ (resp. $K_{n+1}$). Here, a feasible edge-replacement removes an edge and reinserts it so that the resulting graph is isomorphic to $G$; the induced relabeling yields a class of permutations of the label set. Motivated by classical group theoretic ideas, we introduce the hang group, a new invariant that can encode how local amoebas embed into larger ones. Using this framework, we identify necessary and sufficient conditions connecting stem-symmetric and hang-symmetric graphs with local and global amoebas. In particular, we show how hang-symmetry and stem-symmetry conditions propagate under the addition of leaves and isolated vertices, in turn yielding constructive criteria for both local and global amoebas. Finally, via wreath products, we provide four sets of sufficient conditions, one for each property, guaranteeing when the comb product is a local amoeba, a global amoeba, stem-symmetric, or hang-symmetric. These results strengthen and generalize existing constructions of local and global amoebas.

Figures (7)

• Figure 1: Graphs $G$ and $G_{(345)}$ with labels in blue and $L_G = \{13,23,34,45\}$. Notice that $L_G = L_{G_{(345)}}$.
• Figure 2: Diagram of implications related to the global amoeba property
• Figure 3: Diagram of implications related to the local amoeba property
• Figure 4: A graph $G$ that is hang-symmetric with respect to $1$. The sets $\mathcal{E}^1_G = \{ (24)(68), (34)(78), (48), (47) \}$ and $Aut(G)=\{(14)(58)(23)(67)\}$ satisfy that $\langle \mathcal{E}^1_G \cup Aut(G) \rangle = S_8$, but $\langle \mathcal{E}^1_G \rangle \neq S_7$.
• Figure 5: Pictured is the graph $G$ on the top left, the graph $H$ rooted on the red vertex on the top right, and their comb product $G * H$ on the bottom.

Theorems & Definitions (50)

• proof
• Lemma 1: eslava2023new
• Lemma 2: eslava2023new
• Lemma 3: caro2021unavoidable
• Theorem 1: eslava2023new
• Lemma 4
• Proposition 1
• proof : Proof of item $i)$
• Theorem 2
• proof