# Complexity of Modification Problems for Reciprocal Best Match Graphs

## TL;DR

This paper investigates the computational complexity of modifying graphs to become reciprocal best match graphs (RBMGs), showing NP-hardness for general cases and fixed-parameter tractability for 2-colored graphs, with implications for evolutionary biology.

## Contribution

It proves NP-hardness of RBMG modification problems and identifies fixed-parameter tractability for 2-colored graphs, advancing understanding of computational challenges in biological data correction.

## Key findings

- Decision problems for RBMG modification are NP-hard.
- RBMG editing is fixed-parameter tractable for 2-colored graphs.
- Modifying graphs into RBMGs with cograph structure is NP-complete.

## Abstract

Reciprocal best match graphs (RBMGs) are vertex colored graphs whose vertices represent genes and the colors the species where the genes reside. Edges identify pairs of genes that are most closely related with respect to an underlying evolutionary tree. In practical applications this tree is unknown and the edges of the RBMGs are inferred by quantifying sequence similarity. Due to noise in the data, these empirically determined graphs in general violate the condition of being a ``biologically feasible'' RBMG. Therefore, it is of practical interest in computational biology to correct the initial estimate. Here we consider deletion (remove at most $k$ edges) and editing (add or delete at most $k$ edges) problems. We show that the decision version of the deletion and editing problem to obtain RBMGs from vertex colored graphs is NP-hard. Using known results for the so-called bicluster editing, we show that the RBMG editing problem for $2$-colored graphs is fixed-parameter tractable. A restricted class of RBMGs appears in the context of orthology detection. These are cographs with a specific type of vertex coloring known as hierarchical coloring. We show that the decision problem of modifying a vertex-colored graph (either by edge-deletion or editing) into an RBMG with cograph structure or, equivalently, to an hierarchically colored cograph is NP-complete.