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http://dx.doi.org/10.9708/jksci.2015.20.1.227

Maximum Degree Vertex-Based Algorithm for Maximum Clique Problem  

Lee, Sang-Un (Dept. of Multimedia Eng., Gangneung-Wonju National University)
Publication Information
Journal of the Korea Society of Computer and Information / v.20, no.1, 2015 , pp. 227-235 More about this Journal
Abstract
In this paper, I propose a linear time algorithm devised to produce exact solution to NP-complete maximum clique problem. The proposed algorithm firstly, from a given graph G=(V,E), sets vertex $v_i$ of the maximum degree ${\Delta}(G)$ as clique's major vertex. It then selects vertex $v_j$ of ${\Delta}(G)$ among vertices $N_G(v_i)$ that are adjacent to $v_i$, only to determine $N_G(v_i){\cap}N_G(v_j)$ as candidate cliques w and $v_k$. Next it obtains $w=w{\cap}N_G(v_k)$ by sorting $d_G(v_k)$ in the descending order. Lastly, the algorithm executes the same procedure on $G{\backslash}w$ graph to compare newly attained cliques to previously attained cliques so as to choose the lower. With this simple method, multiple independent cliques would also be attainable. When applied to various regular and irregular graphs, the algorithm proposed in this paper has obtained exact solutions to all the given graphs linear time O(n).
Keywords
Maximum clique; Degree; Maximum Degree; Candidate clique;
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