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K-tree

About: K-tree is a research topic. Over the lifetime, 427 publications have been published within this topic receiving 12096 citations.


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TL;DR: It is shown that a complete $k-partite graph is the only graph with clique number $k$ among all degree-equivalent simple graphs, which gives a lower bound on the cliques number, which is sharper than existing bounds on a large family of graphs.
Abstract: In this note, we show that a complete $k$-partite graph is the only graph with clique number $k$ among all degree-equivalent simple graphs. This result gives a lower bound on the clique number, which is sharper than existing bounds on a large family of graphs.

2 citations

Book ChapterDOI
26 Aug 2007
TL;DR: In this paper, polynomial time approximation schemes (PTASs) for the 2-kMRCT and 2-kBVRT problems are presented and a (2 + Ɛ)-approximation algorithm for the2-kBSRT problem is given.
Abstract: In this paper, we investigate some k-tree problems of graphs with given two sources. Let G = (V, E, ω) be an undirected graph with nonnegative edge lengths and two sources s1, s2∈V. The first problem is the 2-source minimum routing cost k-tree (2-kMRCT) problem, in which we want to find a tree T= (VT, ET) spanning kvertices such that the total distance from all vertex in VT to the two sources is minimized, i.e., we want to minimize Σv∈VT{dT(s1, v) + dT(s2, v)}, in which dT(s, v) is the length of the path between s and v on T. The second problem is the 2- source bottleneck source routing cost k-tree (2-kBSRT) problem, in which the objective function is the maximum total distance from any source to all vertices in VT, i.e., maxs∈(s1, s2){Σv∈VT dT(s, v)}. The third problem is the 2-source bottleneck vertex routing cost k-tree (2-kBVRT) problem, in which the objective function is the maximum total distance from any vertex in VT to the two sources, i.e., maxv∈VT{dT(s1, v) + dT(s2, v)}. In this paper, we present polynomial time approximation schemes (PTASs) for the 2-kMRCT and 2-kBVRT problems. For the 2-kBSRT problem, we give a (2 + Ɛ)-approximation algorithm for any Ɛ > 0.

2 citations

Journal ArticleDOI
TL;DR: In this article, the authors generalized chordal graphs to the graphs that have clique representations that are 2-trees, or even series-parallel graphs (partial 2-tree) or maximal outerplanar graphs.
Abstract: Much of the theory—and the applicability—of chordal graphs is based on their being the graphs that have clique trees. Chordal graphs can be generalized to the graphs that have clique representations that are 2-trees, or even series-parallel graphs (partial 2-trees) or outerplanar or maximal outerplanar graphs. The resulting graph classes can be characterized by forbidding contractions of a few induced subgraphs. There is also a plausible application of such graphs to systems biology.

2 citations

Journal ArticleDOI
TL;DR: The special case of the inequalities, where all cycles intersect in two nodes, is considered, and conditions under which these inequalities induce facets of node-capacitated multicut poly topes and bisection cut polytopes are established.

2 citations

Journal ArticleDOI
TL;DR: A tight lower bound is established on the signed clique-transversal number for a regular graph with clique number at most 4 and the decision problem corresponding to the problem of computing is NP-complete even when restricted to doubly chordal graphs.
Abstract: A function f: V→{-1,+1}, defined on the vertices of a graph G, is a signed clique-transversal function (SCTF) if ∑u∈V(C)f(u)≥1 for every clique C of G. The weight of an SCTF is w(f)=∑v∈V(G)f(v). The signed clique-transversal number, denoted [image omitted] , is the minimum weight of an SCTF of G. The signed clique-transversal problem is to find an SCTF of minimum weight for G. In this paper, we establish a tight lower bound on the signed clique-transversal number for a regular graph with clique number at most 4. Furthermore, we show that the decision problem corresponding to the problem of computing [image omitted] is NP-complete even when restricted to doubly chordal graphs. Also, we prove that the signed clique-transversal problem can be solved in linear time for a strongly chordal graph if its strong elimination ordering is given.

2 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20214
20201
20193
20183
201724
201626