Given an undirected graph \(G = (V,E)\) and a positive integer k, a k-vertex-critical subgraph (k-VCS) of G is a subgraph H such that its chromatic number equals k (i.e., \(\chi (H) = k\)), and removing any vertex causes a decrease of \(\chi (H)\). The k-VCS problem (k-VCSP) is to find the smallest k-vertex-critical subgraph \(H^*\) of G. This paper proposes an iterated backtrack-based removal (IBR) heuristic to find k-VCS for a given graph G. IBR extends the popular removal strategy that is intensification-oriented. The proposed extensions include two new diversification-oriented search components—a backtracking mechanism to reconsider some removed vertices and a perturbation strategy to escape local optima traps. Computational results on 80 benchmark graphs show that IBR is very competitive in terms of solution quality and run-time efficiency compared with state-of-the-art algorithms in the literature. Specifically, IBR improves the best-known solutions for 9 graphs and matches the best results for other 70 instances. We investigate the interest of the key components of the proposed algorithm.
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Sun, W., Hao, JK. & Caminada, A. Iterated backtrack removal search for finding k-vertex-critical subgraphs. J Heuristics 25, 565–590 (2019). https://doi.org/10.1007/s10732-017-9358-5
- Vertex-critical subgraph
- Graph coloring
- Tabu search
- Backtracking-based diversification
- Irreducibly inconsistent systems