Determining the Complexity of Chromatic Sum in Classes Defined by a Set of Forbidden Graphs

2026-06-30

AI Summary

This study investigates the computational complexity of the Chromatic Sum problem across various hereditary graph classes defined by forbidden substructures—such as graphs excluding a fixed graph H as a subgraph, induced subgraph, or minor. By developing a novel reduction framework tailored to planar graphs and graphs with bounded independence number, and by integrating structural insights from clique-width theory and forbidden subgraph characterizations, the work establishes an almost complete complexity classification of the problem over these graph families. A key contribution is the proof that Chromatic Sum remains NP-complete on graphs of clique-width at most 3, thereby complementing the known polynomial-time solvability for clique-width at most 2 and delineating a sharp complexity boundary for the problem within broad graph classes.

Abstract

The Chromatic Sum problem asks, given a graph $G$ and an integer $k$, whether $G$ admits a colouring $c$ with sum $\sum_{v\in V}c(v) \leq k$. We study the complexity of Chromatic Sum on graph classes defined by some set of forbidden graphs. First, we show that three known frameworks fully classify the complexity of Chromatic Sum on $HH$-minor-free graphs and $HH$-topological-minor-free graphs for any set of graphs $HH$, and on $HH$-subgraph-free graphs for any finite set of graphs $HH$. To show this, we prove a new NP-completeness result for Chromatic Sum on certain subdivisions of planar subcubic graphs. Next, we consider other containment relations. We formalise a novel framework of problems that are NP-complete for planar graphs as well as for graphs of bounded independence number. For every problem in this framework, we obtain an almost complete complexity classification on $H$-induced-minor-free graphs, $H$-induced-topological-minor-free graphs, and $H$-free graphs for every graph $H$. We show that Chromatic Sum belongs to this framework, as do several other problems. We also define a more fine-grained framework for the induced subgraph relation. We apply this to obtain a complete complexity classification for Chromatic Sum on $H$-free graphs, as well as for several other problems. We justify the choice of this framework by proving that Chromatic Sum is NP-complete for graphs of clique-width at most $3$. This result complements a known polynomial-time result for graphs of clique-width at most $2$.

Problem

• Chromatic Sum
• forbidden graphs
• computational complexity
• graph classes
• NP-completeness

Innovation

• Chromatic Sum
• forbidden graphs
• complexity classification
• clique-width
• NP-completeness