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A tight analysis of the maximal matching heuristic
 IN PROC. OF THE ELEVENTH INTERNATIONAL COMPUTING AND COMBINATORICS CONFERENCE (COCOON), LNCS
, 2005
"... We study the algorithm that iteratively removes adjacent vertices from a simple, undirected graph until no edge remains. This algorithm is a wellknown 2approximation to three classical NPhard optimization problems: MINIMUM VERTEX COVER, MINIMUM MAXIMAL MATCHING and MINIMUM EDGE DOMINATING SET. W ..."
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We study the algorithm that iteratively removes adjacent vertices from a simple, undirected graph until no edge remains. This algorithm is a wellknown 2approximation to three classical NPhard optimization problems: MINIMUM VERTEX COVER, MINIMUM MAXIMAL MATCHING and MINIMUM EDGE DOMINATING SET. We show that the worstcase approximation factor of this simple method can be expressed in a finer way when assumptions on the density of the graph is made. For graphs with an average degree at least ɛn, called weakly ɛdense graphs, we show that the asymptotic approximation factor is min{2, 1/(1 − √ 1 − ɛ)}. For graphs with a minimum degree at least ɛn – strongly ɛdense graphs – we show that the asymptotic approximation factor is min{2, 1/ɛ}. These bounds are obtained through a careful analysis of the tight examples.
Fibonacci Index and Stability Number of Graphs: a Polyhedral Study
, 2008
"... Abstract. The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and conne ..."
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Abstract. The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Turán graphs frequently appear in extremal graph theory. We show that Turán graphs and a connected variant of them are also extremal for these particular problems. We also make a polyhedral study by establishing all the optimal linear inequalities for the stability number and the Fibonacci index, inside the classes of general and connected graphs of order n.
Turán’s Theorem and kConnected Graphs
"... The minimum size of a kconnected graph with given order and stability number is investigated. If no connectivity is required, the answer is given by Turán’s Theorem. For connected graphs, the problem has been solved recently independently by Christophe et al., and by Gitler and Valencia. In this pa ..."
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The minimum size of a kconnected graph with given order and stability number is investigated. If no connectivity is required, the answer is given by Turán’s Theorem. For connected graphs, the problem has been solved recently independently by Christophe et al., and by Gitler and Valencia. In this paper, we give a short proof of their result and determine the extremal graphs. We settle the case of 2connected graphs, characterize the corresponding extremal graphs, and also extend a result of Brouwer related to Turán’s Theorem.
Turán Graphs, Stability Number, and Fibonacci Index
, 2008
"... Abstract. The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and conne ..."
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Abstract. The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Turán graphs frequently appear in extremal graph theory. We show that Turán graphs and a connected variant of them are also extremal for these particular problems.