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A polynomial algorithm for constructing the clique graph of a line graph. (English) Zbl 0592.05034
Summary: We will consider only finite, undirected, connected graphs without loops or multiple edges. A clique of graph G is a maximal complete subgraph of G. The clique graph K(G) of graph G is the intersection graph of the cliques of G. The line graph L(G) of graph G is the intersection graph of the edges of G. Thus, K(L(G)) is the clique graph of the line graph of G. This paper presents a five-step algorithm for constructing K(L(G)). As applications of the algorithm the results of two previous papers on K(L(G)) are subsumed as easy corollaries. Finally, additional corollaries are presented, including a description of \(K(L(K_ n))\). This algorithm is on the order \(O(n^ 3)\), whereas constructing the clique graph of a general graph is NP-complete.
MSC:
05C35 Extremal problems in graph theory
05C99 Graph theory
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References:
[1] Harary, F., Graph theory, (1969), Addison-Wesley Reading, MA · Zbl 0797.05064
[2] Hedetniemi, S.T.; Slater, P.J., Line graphs of triangleless and iterated clique graphs, (), 139-147 · Zbl 0255.05121
[3] Johnson, H.D., Cliques of a graph, Internat. J. comput. information sci., 5, 3, 209-238, (1976) · Zbl 0401.68042
[4] Sato, I., On a few clique graph equations, TRU mathematics, 14, 2, 31-36, (1978) · Zbl 0418.05043
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