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On Generalizations of Pairwise Compatibility Graphs

Tiziana Calamoneri, Manuel Lafond, Angelo Monti, Blerina Sinaimeri

TL;DR

This work introduces two natural generalizations of $PCG$, namely $k$-OR-PCG and $k$-AND-PCG, formalizing them as unions and intersections of $k$ PCGs and linking them to covering number and intersection dimension. It situates these classes with respect to $k$-interval-PCG and other graph families, establishing upper bounds on the minimum $k$ needed for arbitrary graphs and special classes, while proving that no finite $k$ makes $k$-interval-PCG universal through explicit constructions and Ramsey-type arguments. The paper further analyzes multi-interval-PCGs, showing that every graph with $m$ edges is $m$-interval-PCG and that for any $k\, ext{≥}2$ there exist bipartite graphs not in $k$-interval-PCG, implying no universal $k$. It also connects these generalized PCG classes to planarity, boxicity, and arboricity, offering structural insights and a set of open problems for computational complexity and graph-theoretic characterizations.

Abstract

A graph $G$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree and an interval $I$, such that each leaf of the tree is a vertex of the graph, and there is an edge $\{ x, y \}$ in $G$ if and only if the weight of the path in the tree connecting $x$ and $y$ lies within the interval $I$. Originating in phylogenetics, PCGs are closely connected to important graph classes like leaf-powers and multi-threshold graphs, widely applied in bioinformatics, especially in understanding evolutionary processes. In this paper we introduce two natural generalizations of the PCG class, namely $k$-OR-PCG and $k$-AND-PCG, which are the classes of graphs that can be expressed as union and intersection, respectively, of $k$ PCGs. These classes can be also described using the concepts of the covering number and the intersection dimension of a graph in relation to the PCG class. We investigate how the classes of OR-PCG and AND-PCG are related to PCGs, $k$-interval-PCGs and other graph classes known in the literature. In particular, we provide upper bounds on the minimum $k$ for which an arbitrary graph $G$ belongs to $k$-interval-PCGs, $k$-OR-PCG or $k$-AND-PCG classes. For particular graph classes we improve these general bounds. Moreover, we show that, for every integer $k$, there exists a bipartite graph that is not in the $k$-interval-PCGs class, proving that there is no finite $k$ for which the $k$-interval-PCG class contains all the graphs. This answers an open question of Ahmed and Rahman from 2017. Finally, using a Ramsey theory argument, we show that for any $k$, there exists graphs that are not in $k$-AND-PCG, and graphs that are not in $k$-OR-PCG.

On Generalizations of Pairwise Compatibility Graphs

TL;DR

This work introduces two natural generalizations of , namely -OR-PCG and -AND-PCG, formalizing them as unions and intersections of PCGs and linking them to covering number and intersection dimension. It situates these classes with respect to -interval-PCG and other graph families, establishing upper bounds on the minimum needed for arbitrary graphs and special classes, while proving that no finite makes -interval-PCG universal through explicit constructions and Ramsey-type arguments. The paper further analyzes multi-interval-PCGs, showing that every graph with edges is -interval-PCG and that for any there exist bipartite graphs not in -interval-PCG, implying no universal . It also connects these generalized PCG classes to planarity, boxicity, and arboricity, offering structural insights and a set of open problems for computational complexity and graph-theoretic characterizations.

Abstract

A graph is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree and an interval , such that each leaf of the tree is a vertex of the graph, and there is an edge in if and only if the weight of the path in the tree connecting and lies within the interval . Originating in phylogenetics, PCGs are closely connected to important graph classes like leaf-powers and multi-threshold graphs, widely applied in bioinformatics, especially in understanding evolutionary processes. In this paper we introduce two natural generalizations of the PCG class, namely -OR-PCG and -AND-PCG, which are the classes of graphs that can be expressed as union and intersection, respectively, of PCGs. These classes can be also described using the concepts of the covering number and the intersection dimension of a graph in relation to the PCG class. We investigate how the classes of OR-PCG and AND-PCG are related to PCGs, -interval-PCGs and other graph classes known in the literature. In particular, we provide upper bounds on the minimum for which an arbitrary graph belongs to -interval-PCGs, -OR-PCG or -AND-PCG classes. For particular graph classes we improve these general bounds. Moreover, we show that, for every integer , there exists a bipartite graph that is not in the -interval-PCGs class, proving that there is no finite for which the -interval-PCG class contains all the graphs. This answers an open question of Ahmed and Rahman from 2017. Finally, using a Ramsey theory argument, we show that for any , there exists graphs that are not in -AND-PCG, and graphs that are not in -OR-PCG.
Paper Structure (8 sections, 25 theorems, 5 equations, 6 figures)

This paper contains 8 sections, 25 theorems, 5 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Results and organization of the paper.
  3. Preliminaries
  4. Two new generalizations of PCGs
  5. On multi-interval-PCGs
  6. On OR-PCGs
  7. On AND-PCGs
  8. Discussion and open problems

Key Result

Lemma 1

Let $T$ be a tree, and $u$, $v$ and $w$ be three leaves of $T$ such that $P_T(u,v)$ is the longest path in $T_{\{ u, v, w \}}$. Let $x$ be a leaf of $T$ other than $u$, $v$ and $w$. Then, $d_T(x,w) \leq \max\{ d_T (x,u), d_T (x,v)\}$.

Figures (6)

  • Figure 1: (a) A graph G which is not a PCG DMR15. (b) a tree $T$ such that $G=$$2$-interval-PCG$(T,I_1, I_2)$ where $I_1=[1,3]$ and $I_2=[5,6]$.
  • Figure 2: (a) A graph $G$ which is not a PCG BCMP19. The two graphs on the same set of vertices $G_1$ (induced by the double-lined edges) and $G_2$ (induced by the continuous-lined edges) are in PCG, and we provide: (b) a tree $T_1$ and an interval $I_1=[6,10]$ such that $G_1=$ PCG$(T_1,I_1)$; (c) a tree $T_2$ and an interval $I_2=[5,6]$ such that $G_2=$ PCG$(T_2,I_2)$. It clearly holds that $G=$$2$-OR-PCG$(T_1, T_2, I_1, I_2)$.
  • Figure 3: (a) A graph $G$ (considering only the black and continuous-lined edges) which is not a PCG BCMP19, and three sets of edges: the set $E_1$ of blue and dotted edges, the set $E_2$ of red and double-lined edges, and the set $E_3$ of green and long-dashed edges. The three graphs $G_1=K_8 \setminus E_1$, $G_2=K_8 \setminus E_2$, and $G_3=K_8 \setminus E_3$ are in PCG and we provide: (b) a tree $T_1$ such that $G_1=$ PCG$(T_1,I_1)$; (c) a tree $T_2$ such that $G_2=$ PCG$(T_2,I_2)$; (d) a tree $T_3$ such that $G_3=$ PCG$(T_3,I_3)$, where $I_1=I_2=[2,103]$, and $I_3=[3,5]$. It clearly holds that $G=$$3$-AND-PCG$(T_1, T_2, I_1, I_2, I_3)$.
  • Figure 4: Graph $G_3$.
  • Figure 5: The structure of graph $G_i$ in the proof of Theorem \ref{['theo:or_general']}.
  • ...and 1 more figures

Theorems & Definitions (49)

  • Lemma 1: YBR10
  • Definition 1: KPM03
  • Definition 2: NRT02
  • Definition 3: CP12
  • Definition 4: multiintervalPCG
  • Definition 5: $k$-OR-PCG
  • Definition 6: $k$-AND-PCG
  • Theorem 1
  • proof
  • Lemma 2
  • ...and 39 more