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Algorithmic Applications of Tyshkevich's Graph Decomposition: A Primer and a Toolkit

Christine T. Cheng, Chelsea Ann Lambert

TL;DR

Tyshkevich's canonical decomposition characterizes every graph as a composition of indecomposable components, and unigraphs correspond to those whose indecomposable pieces are themselves unigraphs. The paper provides a complete classification of indecomposable unigraphs (split and non-split) and proves that many classical graph parameters (e.g., $\omega(G)$, $\alpha(G)$, $\beta(G)$, $\chi(G)$) and symmetry measures (distinguishing and fixing numbers) can be computed in linear time by aggregating contributions from components, plus a small base graph $G_0$. It further develops a compact decomposition to handle automorphisms efficiently and introduces a toolkit implementing these constructions, enabling generation, recognition, and exact computation for unigraphs. The approach promises efficient exact algorithms for a broad class of problems and motivates extending the methodology to other graph families.

Abstract

A graph that is completely determined by its degree sequence is called a unigraph. In 2000, Regina Tyshkevich published one of the most important papers on unigraphs. There are two parts to the paper: a decomposition theorem that describes how every graph can be broken into a sequence of basic graphs and a complete classification of all basic unigraphs. Together, they reveal how every unigraph is constructed. We provide an informal overview of Tyshkevich's results and show how they enable the computation of various graph parameters of unigraphs in linear time. We also created a toolkit (https://chelseal11.github.io/tyshkevich_decomposition_toolkit/) that implements the algorithms described in this write-up.

Algorithmic Applications of Tyshkevich's Graph Decomposition: A Primer and a Toolkit

TL;DR

Tyshkevich's canonical decomposition characterizes every graph as a composition of indecomposable components, and unigraphs correspond to those whose indecomposable pieces are themselves unigraphs. The paper provides a complete classification of indecomposable unigraphs (split and non-split) and proves that many classical graph parameters (e.g., , , , ) and symmetry measures (distinguishing and fixing numbers) can be computed in linear time by aggregating contributions from components, plus a small base graph . It further develops a compact decomposition to handle automorphisms efficiently and introduces a toolkit implementing these constructions, enabling generation, recognition, and exact computation for unigraphs. The approach promises efficient exact algorithms for a broad class of problems and motivates extending the methodology to other graph families.

Abstract

A graph that is completely determined by its degree sequence is called a unigraph. In 2000, Regina Tyshkevich published one of the most important papers on unigraphs. There are two parts to the paper: a decomposition theorem that describes how every graph can be broken into a sequence of basic graphs and a complete classification of all basic unigraphs. Together, they reveal how every unigraph is constructed. We provide an informal overview of Tyshkevich's results and show how they enable the computation of various graph parameters of unigraphs in linear time. We also created a toolkit (https://chelseal11.github.io/tyshkevich_decomposition_toolkit/) that implements the algorithms described in this write-up.
Paper Structure (7 sections, 25 theorems, 8 equations, 4 figures, 4 tables, 3 algorithms)

This paper contains 7 sections, 25 theorems, 8 equations, 4 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Decomposition Theorem
  3. Indecomposable Unigraphs
  4. Applications, Part 1
  5. Applications, Part 2
  6. A Toolkit
  7. Conclusion

Key Result

Theorem 2.1

HaSi81 For any $KS$-partition $(A,B)$ of split graph $G$, exactly one of the following holds: where $\omega(G)$ and $\alpha(G)$ are the clique and independence numbers of $G$ respectively. Moreover, in a $K$-max partition, there is a vertex $u \in A$ so that $B \cup \{u\}$ induces a stable set while in an $S$-max partition, there is a vertex $v \in B$ so that $A \cup \{v\}$ induces a complet

Figures (4)

  • Figure 1: The tree $T$ on the left has degree sequence $(3,2,1,1,1) = (3, 2, 1^3)$. It has one $KS$-partition $(\{b,d\}, \{a,c,e\})$. It is combined with the another graph $H$, a $3$-cycle, using the composition operation to create the graph $(T, A, B) \circ H$.
  • Figure 2: The threshold graph $(\{v_4\}, \{v_4\}, \emptyset) \circ (\{v_3\}, \{v_3\}, \emptyset) \circ (\{v_2\}, \emptyset, \{v_2\}) \circ (\{v_1\}, \emptyset, \{v_1\}) \circ \{v_0 \}$.
  • Figure 3: Given the tree $T$ on the left with $A = \{b,d\}$ and $B = \{a, c, e\}$, the complement, the inverse and the inverse of the complement of $(T, A, B)$ are shown on the right.
  • Figure 4: A distinguishing labeling of $C_5$ on the left and a fixing set of $C_5$ on the right. Vertices that are part of the fixing set are assigned distinct colors and those not in the set are assigned a null color.

Theorems & Definitions (33)

  • Theorem 2.1
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • ...and 23 more