[Paper Review] Solving the clique cover problem on (bull, $C_4$)-free graphs

Kathie Cameron, Chı́nh T. Hoàng|arXiv (Cornell University)|Apr 2, 2017

Advanced Graph Theory Research12 references4 citations

TL;DR

This paper presents an O(n⁴) polynomial-time algorithm to compute the minimum clique cover for (bull, C₄)-free graphs, leveraging structural decomposition via terminal one-point cutsets and reducibility. The key contribution is solving the clique cover problem in polynomial time for this class, which is equivalent to 3-coloring in the complement class (bull, 2K₂)-free graphs.

ABSTRACT

We give an $O(n^4)$ algorithm to find a minimum clique cover of a (bull, $C_4$)-free graph, or equivalently, a minimum colouring of a (bull, $2K_2$)-free graph, where $n$ is the number of vertices of the graphs.

Motivation & Objective

• To develop a polynomial-time algorithm for the minimum clique cover problem in (bull, C₄)-free graphs.
• To extend known results on chordal and basic graphs to a broader class of graphs with forbidden induced subgraphs.
• To address the open problem of clique cover complexity in (bull, house)-free and cap-free graphs by analyzing structural properties.
• To provide a constructive algorithm based on vertex domination, one-point cutsets, and matching-based clique cover in triangle-free subgraphs.
• To contribute toward the long-standing open problem of polynomial-time coloring for AT-free graphs by solving a key subclass.

Proposed method

• Use vertex domination reduction: if vertex x dominates y, remove x without changing the clique cover number.
• Apply structural decomposition: if the graph is irreducible and not basic, it must contain a terminal one-point cutset.
• For a terminal one-point cutset v, analyze components Gᵢ = G[Cᵢ ∪ {v}] where Gᵢ is triangle-free.
• Compute maximum matching m(Gᵢ) and m(Gᵢ − v) to determine whether v can be covered as a singleton or must be paired.
• Use recursive decomposition: if m(Gᵢ) = m(Gᵢ − v), then θ(G) = θ(G′) + θ(Gᵢ − v); otherwise θ(Gᵢ) = θ(Gᵢ − v).
• Combine results via dynamic recursion over at most n vertices, ensuring O(n⁴) time complexity via efficient matching and cutset detection.

Experimental results

Research questions

• RQ1 Can the minimum clique cover problem be solved in polynomial time for (bull, C₄)-free graphs?
• RQ2 What structural properties of (bull, C₄)-free graphs enable a polynomial-time solution to the clique cover problem?
• RQ3 How do terminal one-point cutsets and reducibility influence the design of a recursive clique cover algorithm?
• RQ4 To what extent does this algorithm contribute to the broader open problem of coloring AT-free graphs?
• RQ5 Is the clique cover problem tractable for (bull, house)-free or cap-free graphs, given this result?

Key findings

• An O(n⁴) algorithm is presented for computing the minimum clique cover in (bull, C₄)-free graphs.
• The algorithm reduces the problem using vertex domination and recursively decomposes the graph via terminal one-point cutsets.
• For each component Gᵢ induced by a terminal one-point cutset v, the algorithm checks whether v can be covered as a singleton or must be in a size-2 clique via matching comparison.
• The time complexity is bounded by O(n⁴), derived from O(n) recursive steps, each involving O(nm) cutset checks and O(√n m) matching computations.
• The result implies that the chromatic number of (bull, 2K₂)-free graphs can be computed in O(n⁴) time, as it is equivalent to the clique cover problem in the complement.
• The algorithm confirms that the clique cover problem is polynomial-time solvable for this class, contributing to the open problem of coloring AT-free graphs.