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Strongly chordal graphs as intersection graphs of trees (Farber's proof revisited)

Therese Biedl

TL;DR

The paper addresses the problem of characterizing strongly chordal graphs via geometric representations. It formalizes a characterization: a graph $G$ is strongly chordal iff it is the intersection graph of a compatible collection of subtrees of a rooted weighted tree, a result attributed to Farber’s thesis. It then provides an accessible reconstruction of the proof, including an alternate sufficiency argument that converts a compatible tree-representation into a strong elimination order, achievable in $O(n+m)$ time given the overshadow relations for edges. The work clarifies the structural role of the overshadows relation, introduces an efficient linear-time algorithm to extract strong elimination orders, and discusses open questions about unweighted representations and unit-length trees, with practical impact on algorithmic applications for strongly chordal graphs.

Abstract

In his Ph.D. thesis, Farber proved that every strongly chordal graph can be represented as intersection graph of subtrees of a weighted tree, and these subtrees are ``compatible''. Moreover, this is an equivalent characterization of strongly chordal graphs. To my knowledge, Farber never published his results in a conference or a journal, and the thesis is not available electronically. As a service to the community, I therefore reproduce the proof here. I then answer some questions that naturally arise from the proof. In particular, the sufficiency proof works by showing the existence of a simple vertex. I give here an alternate sufficiency proof that directly converts a set of compatible subtrees into a strong elimination order.

Strongly chordal graphs as intersection graphs of trees (Farber's proof revisited)

TL;DR

The paper addresses the problem of characterizing strongly chordal graphs via geometric representations. It formalizes a characterization: a graph is strongly chordal iff it is the intersection graph of a compatible collection of subtrees of a rooted weighted tree, a result attributed to Farber’s thesis. It then provides an accessible reconstruction of the proof, including an alternate sufficiency argument that converts a compatible tree-representation into a strong elimination order, achievable in time given the overshadow relations for edges. The work clarifies the structural role of the overshadows relation, introduces an efficient linear-time algorithm to extract strong elimination orders, and discusses open questions about unweighted representations and unit-length trees, with practical impact on algorithmic applications for strongly chordal graphs.

Abstract

In his Ph.D. thesis, Farber proved that every strongly chordal graph can be represented as intersection graph of subtrees of a weighted tree, and these subtrees are ``compatible''. Moreover, this is an equivalent characterization of strongly chordal graphs. To my knowledge, Farber never published his results in a conference or a journal, and the thesis is not available electronically. As a service to the community, I therefore reproduce the proof here. I then answer some questions that naturally arise from the proof. In particular, the sufficiency proof works by showing the existence of a simple vertex. I give here an alternate sufficiency proof that directly converts a set of compatible subtrees into a strong elimination order.

Paper Structure

This paper contains 10 sections, 9 theorems, 13 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representations of strongly chordal graphs
  4. Discussion items
  5. The $\mathop{\mathrm{\raisebox{-0ex}{!}\raisebox{-0ex}{!} }}\nolimits$ relationship
  6. Obtaining strong elimination orders
  7. Unweighted tree representations?
  8. The original proof
  9. Preliminaries
  10. The actual proof

Key Result

Theorem 1

A graph is strongly chordal if and only if it is the intersection graph of a compatible collection of subtrees of a rooted weighted tree.

Figures (3)

  • Figure 1: (a) A chordal graph (actually strongly chordal) with a perfect elimination order (actually a strong elimination order). (b) The tree-representation that we get when applying Lemma \ref{['lem:order_treerep']}. Symbol means that this is node $\mathbf{x}$ and it belongs to $T(x),T(y)$ and $T(z)$. Dotted lines indicate cutoff values for some edges; we also indicate some witnesses for not overshadowing. (c) Abstract illustration of the concept of overshadowing.
  • Figure 2: (a) Three compatible subtrees where $T_1\mathop{\mathrm{\raisebox{-0ex}{!}\raisebox{-0ex}{!} }}\nolimits T_2 \mathop{\mathrm{\raisebox{-0ex}{!}\raisebox{-0ex}{!} }}\nolimits T_3$, but $T_1\mathop{\mathrm{\raisebox{-0ex}{!}\raisebox{-0ex}{!!} }}\nolimits T_3$. (b) Three compatible intersecting subtrees $T_1\mathop{\mathrm{\raisebox{-0ex}{!}\raisebox{-0ex}{!} }}\nolimits T_2 \mathop{\mathrm{\raisebox{-0ex}{!}\raisebox{-0ex}{!} }}\nolimits T_3 \mathop{\mathrm{\raisebox{-0ex}{!}\raisebox{-0ex}{!} }}\nolimits T_1$. (c) For the proof of Lemma \ref{['lem:not_cyclic']}.
  • Figure 3: A tree representation that is compatible (with $T(k)\mathop{\mathrm{\raisebox{-0ex}{!}\raisebox{-0ex}{!} }}\nolimits T(\ell)$). One possible bottom-up enumeration order is $i,j,k,\ell$, which is not a strong elimination order since there are edges $(i,k),(i,\ell)$ and $(j,k)$, but no edge $(j,\ell)$.

Theorems & Definitions (18)

  • Theorem 1: FarberThesis
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • proof
  • ...and 8 more