Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Characterization of Split Comparability Graphs

Tithi Dwary, Khyodeno Mozhui, K. V. Krishna

TL;DR

The paper addresses characterizing split comparability graphs by a labeling of the clique $C$ and partitioning the independent set $I$ into $I_1,I_2,I_3$, establishing a transitive orientation criterion that is both necessary and sufficient. It further provides a constructive procedure to obtain a $3$-uniform permutational representation, proving $\,\mathcal{R}^p(G)\le 3$, with equality precisely when the induced subgraph $B_4$ is present, thereby linking graph representations to the dimension of induced posets. The results yield an alternative proof that the dimension of a split order is at most $3$ and identify the condition for tightness via $B_4$. The work blends structural labeling with algorithmic construction to show that every split comparability graph admits a $3$-uniform permutational representation, offering both theoretical insights and a practical representation method.

Abstract

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A split comparability graph is a split graph which is transitively orientable. In this work, we characterize split comparability graphs in terms of vertex labelling. Further, using this characterization, we prove that the permutation-representation number of a split comparability graph is at most three. This gives us an alternative proof of the result in order theory that the dimension of a split order is at most three.

Characterization of Split Comparability Graphs

TL;DR

The paper addresses characterizing split comparability graphs by a labeling of the clique and partitioning the independent set into , establishing a transitive orientation criterion that is both necessary and sufficient. It further provides a constructive procedure to obtain a -uniform permutational representation, proving , with equality precisely when the induced subgraph is present, thereby linking graph representations to the dimension of induced posets. The results yield an alternative proof that the dimension of a split order is at most and identify the condition for tightness via . The work blends structural labeling with algorithmic construction to show that every split comparability graph admits a -uniform permutational representation, offering both theoretical insights and a practical representation method.

Abstract

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A split comparability graph is a split graph which is transitively orientable. In this work, we characterize split comparability graphs in terms of vertex labelling. Further, using this characterization, we prove that the permutation-representation number of a split comparability graph is at most three. This gives us an alternative proof of the result in order theory that the dimension of a split order is at most three.
Paper Structure (7 sections, 12 theorems, 1 equation, 1 figure, 1 algorithm)

This paper contains 7 sections, 12 theorems, 1 equation, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction and Preliminaries
  2. Our Contributions
  3. Characterization
  4. Claim: $a, c \not\in A_3$.
  5. Claim: $a, c \not\in A_2$.
  6. Claim: $a, c \not\in A_1$.
  7. Permutation-Representation Number

Key Result

theorem 1

Let $G$ be a split graph. Then, $G$ is a permutation graph if and only if $G$ is a $\mathcal{B}$-free graph, where $\mathcal{B}$ is the class of graphs given in Fig. fig7.

Figures (1)

  • Figure 1: The family of graphs $\mathcal{B}$

Theorems & Definitions (22)

  • theorem 1: Split_circle_graphs
  • theorem 2: Golumbicbook_2004
  • theorem 3: Kitaev_2021
  • theorem 4: Kitaev_2021Kitaev_2024
  • lemma 1
  • proof
  • lemma 2
  • proof
  • theorem 5
  • proof
  • ...and 12 more