Recognition Algorithm for Diamond-Free Graphs

The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique cover.

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“…Algorithm 1: Weak decomposition of a graph [23] Input: G = (V, E) connected graph that have two or more nonadjacent vertices.…”

Section: Methodsmentioning

confidence: 99%

“…[22,23]). Any incomplete and connected graph G = (V, E) admits a weak component; let us denote it with A, such that G(V − A) = G(N(A)) + G(N(A)).Theorem 9 ([24,25]).…”

mentioning

confidence: 99%

Recognition and Optimization Algorithms for P5-Free Graphs

et al. 2020

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“…A diamond is the graph that results from a four-vertex clique by deleting one edge. Diamond-free graphs, that is, graphs containing no diamond as an induced subgraph, are a natural graph class and have been already studied in earlier work [2,34]. Proposition 1.…”

Section: Definition 1 (S-vertex-overlap Property and S-edge-overlap Pmentioning

confidence: 99%

Graph-based data clustering with overlaps

Fellows

Guo

Komusiewicz

et al. 2011

Discrete Optimization

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We introduce overlap cluster graph modification problems where, other than in most previous work, the clusters of the target graph may overlap. More precisely, the studied graph problems ask for a minimum number of edge modifications such that the resulting graph consists of clusters (that is, maximal cliques) that may overlap up to a certain amount specified by the overlap number s. In the case of s-vertex-overlap, each vertex may be part of at most s maximal cliques; s-edge-overlap is analogously defined in terms of edges. We provide a complexity dichotomy (polynomial-time solvable versus NP-hard) for the underlying edge modification problems, develop forbidden subgraph characterizations of "cluster graphs with overlaps", and study the parameterized complexity in terms of the number of allowed edge modifications, achieving fixed-parameter tractability (in case of constant s-values) and parameterized hardness (in case of unbounded s-values).

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“…However, the fast matrix multiplication can be avoided while improving the time complexity to O(m 3/2 ) time, as it is shown by Eisenbrand and Grandoni [6]. Talmaciu and Nechita [17] devised a recognition algorithm based on decompositions, but they claim that in the worst case the time required by their algorithm is not better than the one by Kloks et al Note that Theorem 9 implies that there is an O(α(G) 2 m) time algorithm for recognizing whether a graph is a diamond-free graph, improving over the previous algorithms for some sparse graphs. Finally, Vassilevska [19] used the algorithm by Eisenbrand and Grandoni to find an induced K k \ e in a graph.…”

Section: Dynamic Recognition Of Diamond-free Graphsmentioning

confidence: 99%

Arboricity, h-Index, and Dynamic Algorithms

Lin,

Soulignac,

Szwarcfiter

2010

Preprint

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In this paper we present a modification of a technique by Chiba and Nishizeki [Chiba and Nishizeki: Arboricity and Subgraph Listing Algorithms, SIAM J. Comput. 14(1), pp. 210-223 (1985)]. Based on it, we design a data structure suitable for dynamic graph algorithms. We employ the data structure to formulate new algorithms for several problems, including counting subgraphs of four vertices, recognition of diamond-free graphs, cop-win graphs and strongly chordal graphs, among others. We improve the time complexity for graphs with low arboricity or h-index.

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Recognition Algorithm for Diamond-Free Graphs

Cited by 9 publications

References 4 publications

Recognition and Optimization Algorithms for P5-Free Graphs

Recognition and Optimization Algorithms for P5-Free Graphs

Graph-based data clustering with overlaps

Arboricity, h-Index, and Dynamic Algorithms

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