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Simpler and Unified Recognition Algorithm for Path Graphs and Directed Path Graphs

Lorenzo Balzotti

TL;DR

The paper addresses the recognition problem for path graphs and directed path graphs, two graph classes situated between interval and chordal graphs. It builds a unified recognition framework by extending the Ab characterization of path graphs to directed path graphs and by using clique-separator decompositions and antipodality graph colorings. The main contribution is a first, simpler $O(p(m+n))$-time recognition algorithm for both PG and DPG, which avoids complex data structures and relies on a coloring-based recursion that translates into a clique path tree. This has practical implications for efficient recognition in graph libraries and related applications, providing a cohesive approach to two closely related graph classes. The work also clarifies the relationship between existing characterizations and yields a streamlined pathway to implementable algorithms for both PG and DPG.

Abstract

A path graph is the intersection graph of paths in a tree. A directed path graph is the intersection graph of paths in a directed tree. Even if path graphs and directed path graphs are characterized very similarly, their recognition algorithms differ widely. We further unify these two graph classes by presenting the first recognition algorithm for both path graphs and directed path graphs. We deeply use a recent characterization of path graphs, and we extend it to directed path graphs. Our algorithm does not require complex data structures and has an easy and intuitive implementation, simplifying recognition algorithms for both graph classes.

Simpler and Unified Recognition Algorithm for Path Graphs and Directed Path Graphs

TL;DR

The paper addresses the recognition problem for path graphs and directed path graphs, two graph classes situated between interval and chordal graphs. It builds a unified recognition framework by extending the Ab characterization of path graphs to directed path graphs and by using clique-separator decompositions and antipodality graph colorings. The main contribution is a first, simpler -time recognition algorithm for both PG and DPG, which avoids complex data structures and relies on a coloring-based recursion that translates into a clique path tree. This has practical implications for efficient recognition in graph libraries and related applications, providing a cohesive approach to two closely related graph classes. The work also clarifies the relationship between existing characterizations and yields a streamlined pathway to implementable algorithms for both PG and DPG.

Abstract

A path graph is the intersection graph of paths in a tree. A directed path graph is the intersection graph of paths in a directed tree. Even if path graphs and directed path graphs are characterized very similarly, their recognition algorithms differ widely. We further unify these two graph classes by presenting the first recognition algorithm for both path graphs and directed path graphs. We deeply use a recent characterization of path graphs, and we extend it to directed path graphs. Our algorithm does not require complex data structures and has an easy and intuitive implementation, simplifying recognition algorithms for both graph classes.

Paper Structure

This paper contains 18 sections, 17 theorems, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Our approach
  4. Organization
  5. Earlier characterizations of path graphs and directed path graphs
  6. A recent characterization of path graphs
  7. Recognition algorithm for path graphs
  8. The algorithm and its correctness
  9. Implementation details and time complexity
  10. Step \ref{['al:chord']} and Step \ref{['al:Q']}
  11. Step \ref{['al:rec']}
  12. Step \ref{['al:inizialization']}
  13. Step \ref{['al:crossing']}
  14. Step \ref{['al:antipodal']}
  15. Step \ref{['al:ct']}
  16. ...and 3 more sections

Key Result

Theorem 1

A graph $G=(V,E)$ is a path graph (resp. directed path graph) if and only if there exists a tree $T$ (resp. directed tree $T$) with vertex set $\mathbf{C}$, such that for every $v\in V$, $T[\mathbf{C}_v]$ is a path (resp. directed path) in $T$.

Figures (3)

  • Figure 1: a path graph $G$ (on the left) and a clique path tree of $G$ (on the right).
  • Figure 2: a directed path graph $G$ (on the left) and a directed clique path tree of $G$ (on the right).
  • Figure 3: a clique separator $C$ in $G$ of Figure \ref{['fig:examples_Path_Graph_and_cpt']}. The connected components $\gamma_1,\ldots,\gamma_5$ are highlighted by different colors and hatchings.

Theorems & Definitions (19)

  • Theorem 1: gavril_DV_algorithmgavril_UV_algorithm
  • Theorem 2: gavril1
  • Theorem 3: mew
  • Theorem 4: mew
  • Theorem 5: chaplick-gutierrez
  • Lemma 1: ab
  • Lemma 2: ab
  • Theorem 6: ab
  • Theorem 7
  • Definition 1: schaffer
  • ...and 9 more