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Packing $$r$$-cliques in weighted chordal graphs. (English) Zbl 1091.90073
Summary: In P. Hell et al. [Discrete Appl. Math. 141, No. 1–3, 185–194 (2004; Zbl 1043.05097)], we have previously observed that, in a chordal graph $$G$$, the maximum number of independent $$r$$-cliques (i.e., of vertex disjoint subgraphs of $$G$$, each isomorphic to $$K_r$$, with no edges joining any two of the subgraphs) equals the minimum number of cliques of $$G$$ that meet all the $$r$$-cliques of $$G$$. When $$r = 1$$, this says that chordal graphs have independence number equal to the clique covering number. When $$r = 2$$, this is equivalent to a result of Cameron (1989), and it implies a well known forbidden subgraph characterization of split graphs. In this paper we consider a weighted version of the above min-max optimization problem. Given a chordal graph $$G$$, with a nonnegative weight for each $$r$$-clique in $$G$$, we seek a set of independent $$r$$-cliques with maximum total weight. We present two algorithms to solve this problem, based on the principle of complementary slackness. The first one operates on a graph derived from $$G$$, and is an adaptation of an algorithm of M. Farber [Oper. Res. Let. 1, 134–138 (1982; Zbl 0495.05053)]. The second one improves the performance by reducing the number of constraints of the linear programs. Both results produce a min-max relation. When the algorithms are specialized to the situation in which all the $$r$$-cliques have the same weight, we simplify the algorithms reported in Hell et al. (loc. cit.), although these simpler algorithms are not as efficient. As a byproduct, we also obtain new elementary proofs of the above min-max result.

MSC:
 90C35 Programming involving graphs or networks 05C85 Graph algorithms (graph-theoretic aspects)
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References:
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