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Local Equivalence Classes of Distance-Hereditary Graphs using Split Decompositions

Nicholas Connolly, Shin Nishio, Kae Nemoto

TL;DR

This work uses a technique known as split decomposition to establish upper bounds on equivalence class sizes, and it is proved these bounds are tight through a combinatorial enumeration of the graphs'decomposed structure up to symmetry.

Abstract

Local complement is a graph operation formalized by Bouchet which replaces the neighborhood of a chosen vertex with its edge-complement. This operation induces an equivalence relation on graphs; determining the size of the resulting equivalence classes is a challenging problem in general. Bouchet obtained formulas only for paths and cycles, and brute-force methods are limited to very small graphs. In this work, we extend these results by deriving explicit formulas for several broad families of distance-hereditary graphs, including complete multipartite graphs, clique-stars, and repeater graphs. Our approach uses a technique known as split decomposition to establish upper bounds on equivalence class sizes, and we prove these bounds are tight through a combinatorial enumeration of the graphs' decomposed structure up to symmetry.

Local Equivalence Classes of Distance-Hereditary Graphs using Split Decompositions

TL;DR

This work uses a technique known as split decomposition to establish upper bounds on equivalence class sizes, and it is proved these bounds are tight through a combinatorial enumeration of the graphs'decomposed structure up to symmetry.

Abstract

Local complement is a graph operation formalized by Bouchet which replaces the neighborhood of a chosen vertex with its edge-complement. This operation induces an equivalence relation on graphs; determining the size of the resulting equivalence classes is a challenging problem in general. Bouchet obtained formulas only for paths and cycles, and brute-force methods are limited to very small graphs. In this work, we extend these results by deriving explicit formulas for several broad families of distance-hereditary graphs, including complete multipartite graphs, clique-stars, and repeater graphs. Our approach uses a technique known as split decomposition to establish upper bounds on equivalence class sizes, and we prove these bounds are tight through a combinatorial enumeration of the graphs' decomposed structure up to symmetry.
Paper Structure (34 sections, 15 theorems, 14 equations, 16 figures, 5 tables, 2 algorithms)

This paper contains 34 sections, 15 theorems, 14 equations, 16 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Connection with Quantum Information Theory
  4. Structure of this Paper
  5. Summary of Special Graph Classes
  6. Local Complements
  7. Local Complement as a Graph Operation
  8. Local Complement as a Function
  9. Local Complement as a Group Action
  10. LC Equivalence Classes and LC Orbits
  11. LC Invariants
  12. Split Decompositions
  13. Splits and Strong Splits
  14. Strong Split Trees and Split Components
  15. The Quotient-Augmented Strong Split Tree
  16. ...and 19 more sections

Key Result

Lemma 5.1

Let $G$ be a graph with a vertex $q\in V(G)$. Let $Q$ be the quotient graph containing $q$ in the split decomposition of $G$. If $G'$ is a graph obtained from $G$ by a one-vertex extension $p$ of $q$, then $QASST(G')$ is computed from $QASST(G)$ by converting $Q$ into $Q'$ according to a number of c

Figures (16)

  • Figure 1: Examples of special classes of graphs, all distance-hereditary except for the cycle.
  • Figure 2: Graphs related via local complement on vertex $1$. The edges between the vertices in $V(N_G(1))=\{2,3,5\}=V(N_{G'}(1))$ are deleted, while missing edges are added.
  • Figure 3: The LC equivalence class of the complete graph on four vertices, the LC orbit ${\mathcal{O}}(K_4)$.
  • Figure 4: Examples showing the basic concepts of splits in a graph, indicated by the red dashed line.
  • Figure 5: Example of a graph with two strong splits. \ref{['fig:Strong_Split_Tree_example_G_and_SST']} A graph $G$ with its strong splits marked by red dashed lines and the corresponding strong split tree $SST(G)$. \ref{['fig:Strong_Split_Tree_example_Quotient_Graphs']} The quotient graphs corresponding to this split decomposition; squares denote split-nodes. \ref{['fig:Strong_Split_Tree_example_induced_splits']} Visualization of the relationship between the two internal edges in $SST(G)$ and the the corresponding strong splits in $G$. Deleting an edge in the tree divides it into two parts; the corresponding bipartition of leaves in the tree defines the split in the graph. Notice that the subgraphs induced by the edges crossing between both halves of the splits are complete bipartite.
  • ...and 11 more figures

Theorems & Definitions (18)

  • Lemma 5.1
  • proof
  • Theorem 5.2: Originally established in hammer1990completely
  • proof
  • Lemma 6.1: Bouchet bouchet1987reducing, Lemma 2.1
  • Corollary 6.2
  • Lemma 6.3: Bouchet bouchet1988transforming, Corollary 3.3
  • Lemma 6.4: Bouchet bouchet1988transforming, Corollary 4.2
  • Lemma 6.5: Bouchet bouchet1987reducing, Lemma 2.2
  • Theorem 6.6: Mulder's Conjecture mulder1986, proven by Bouchet bouchet1988transforming, Corollary 5.4
  • ...and 8 more