Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A characterization of interval nest digraphs

Ayelén Alcantar, Flavia Bonomo, Guillermo Durán, Nina Pardal

Abstract

A digraph consisting of a set of vertices $V$ and a set of arcs $E$ is called an interval digraph if there exists a family of closed intervals $\{I_u,J_u\}_{u \in V}$ such that $uv$ is an arc if and only if the intersection of $I_u$ and $J_v$ is non-empty. Interval digraphs naturally generalize interval graphs, by extending the classical interval intersection model to directed graphs. Several subclasses of interval digraphs have been studied in the literature-such as balanced, chronological and catch interval digraphs-each characterized by admitting interval representations that satisfy specific restrictions. Among these, interval nest digraphs are the ones that admit an interval representation in which $J_u$ is contained in $I_u$ for all vertices $u$ of $V$. In this work, we provide a complete characterization of interval nest digraphs in terms of vertex linear orderings with forbidden patterns, which we call nest orderings. This result completes the picture of vertex-ordering characterizations among the main subclasses of interval digraphs.

A characterization of interval nest digraphs

Abstract

A digraph consisting of a set of vertices and a set of arcs is called an interval digraph if there exists a family of closed intervals such that is an arc if and only if the intersection of and is non-empty. Interval digraphs naturally generalize interval graphs, by extending the classical interval intersection model to directed graphs. Several subclasses of interval digraphs have been studied in the literature-such as balanced, chronological and catch interval digraphs-each characterized by admitting interval representations that satisfy specific restrictions. Among these, interval nest digraphs are the ones that admit an interval representation in which is contained in for all vertices of . In this work, we provide a complete characterization of interval nest digraphs in terms of vertex linear orderings with forbidden patterns, which we call nest orderings. This result completes the picture of vertex-ordering characterizations among the main subclasses of interval digraphs.
Paper Structure (3 sections, 6 theorems, 9 equations, 7 figures)

This paper contains 3 sections, 6 theorems, 9 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Characterization of interval nest digraphs
  3. Conclusions and Future Work

Key Result

Theorem 1

Let $D=(V,E)$ be a finite digraph. Then $D$ is an interval nest digraph if and only if $D$ is reflexive and there exists a nest ordering on $V$.

Figures (7)

  • Figure 1: Forbidden patterns for adjusted interval digraphs (left) and interval catch digraphs (right), where $u < v < w$, extracted from the characterizations in Feder2009 and Mae-dig, respectively.
  • Figure 2: Forbidden patterns characterizing interval point digraphs, where possibly $u= v$ in (p), $v = w$ in (q) and (r), and $w=z$ in (s) S-D-R-W.
  • Figure 3: Forbidden patterns for balanced interval (catch) digraphs, for $u < v < w < z$, extracted from the characterization in sanchita-ghos.
  • Figure 4: Forbidden patterns for chronological interval digraphs, for $u < v < w$, extracted from the characterization in SandipchronID.
  • Figure 5: Forbidden patterns for reflexive interval digraphs, where possibly $v = w$ in (a), (b), (d) and (e) (figure reproduced from Francis2021Kernel). A dashed loop is the seventh forbidden pattern.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • proof
  • Definition 2
  • Definition 3
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 5 more