Recognizing Leaf Powers and Pairwise Compatibility Graphs is NP-Complete

TL;DR

This work resolves a long-standing question by showing that recognizing leaf powers and pairwise compatibility graphs is NP-hard, and extends the hardness to a broad hierarchy of generalized leaf powers GLP$(q)$ for every $q\ge 1$. The authors achieve this via a novel reduction from the Triangle Ordinal Clustering (TOC) problem to GLP recognition, exploiting the four-point condition to enforce tree-metric constraints and constructing polynomial-size gadgets that encode triangle-distance orders. They establish GLP$(q)$ membership in NP with a polynomial-size certificate by bounding weights and tree size, and prove that GLP$(q)$ recognition is NP-hard through the inductive GLP$(q)\rightarrow GLP(q+1)$ reduction. The results imply NP-hardness for leaf powers, PCGs, and multi-interval PCGs, and provide a framework bridging TOC realizability with tree-metric graph classes, with potential implications for phylogenetics and related graph-embedding problems.

Abstract

Leaf powers and pairwise compatibility graphs were introduced over twenty years ago as simplified graph models for phylogenetic trees. Despite significant research, several properties of these graph classes remain poorly understood. In this paper, we establish that the recognition problem for both classes is NP-complete. We extend this hardness result to a broader hierarchy of graph classes, including pairwise compatibility graphs and their generalizations, multi interval pairwise compatibility graphs.

Figures (4)

• Figure 1: If we satisfy the conditions of Lemma \ref{['lem:4-PC-split']}, we must be in this configuration. $x$ must connect to the left of $M_a$, where vertices are closest to $a_1$ and $a_2$ and $y$ connects to the right of $M_b$, where vertices are closest to $b_1$ and $b_2$.
• Figure 2: By the conditions of Lemma \ref{['lem:twins']} the other 3 vertices are all closer to $a_1$ than to $a_2$, so the only place $x$ can be is on the branch connecting $a_2$ to the tree formed by the other 3 leaves.
• Figure 3: Subgraph generated by $i^\pm\in S'$ with $\left\{x,y\right\}\subseteq S'\setminus \left\{i^{\pm}\right\}$ such that $i^\pm x\prec_{S'} i^\pm y$.
• Figure 4: Subgraph generated by $i\in S$ and $x\in S'$. We denote $\mathop{\mathrm{diam}}\nolimits(T')=D$ to simplify the notation of the edge weights. Additionally, the distance between $v_x$ and $p'_i$ in $T$ to be $d$.

Theorems & Definitions (42)

• Theorem 1.0
• Corollary 1.1
• Theorem 1.2
• Theorem 1.2
• Remark
• Claim 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3