Efficient Analy sis of Graphs with Small Minimal Separators

Skodinis, Konstantin

doi:10.1007/3-540-46784-x_16

Cited by 8 publications

(12 citation statements)

References 20 publications

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“…Such graph classes include (amongst others) circular-arc graphs (O (n 2 ) [18]), polygon-circle graphs (O (n 2 ) [22]), weakly triangulated graphs (O (n 2 ) [7]). Furthermore, computing treewidth is fixed-parameter tractable with respect to the maximum size of a minimal separator [21]. This parameter corresponds to the solution size of Maximum Weight Minimal Separator on unweighted graphs.…”

Section: Related Workmentioning

confidence: 99%

On the maximum weight minimal separator

Hanaka

Bodlaender

Zanden

et al. 2019

Theoretical Computer Science

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Section: Related Workmentioning

confidence: 99%

On the maximum weight minimal separator

Hanaka

Bodlaender

Zanden

et al. 2019

Theoretical Computer Science

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“…They include chordal graphs (for which t = 1, since every chordal graphs is the intersection graph of subtrees in a tree [31,74,116]) and circular-arc graphs, that is, intersection graphs of circular arcs on a circle (for which t = 2), as well as H-graphs, that is, the intersection graphs of connected subgraphs of a subdivision of a fixed multigraph H, introduced in 1992 by Bíró, Hujter, and Tuza [14] and studied more recently in a number of papers [36][37][38]67]. Classes of graphs in which all minimal separators are of bounded size were studied in 1999 by Skodinis [107]. In [54,55], the authors characterized, for each of six wellknown graph containment relations (the minor, topological subgraph, subgraph, and their induced variants), the graphs H such that the class of graphs excluding a H with respect to the relation is (tw, ω)-bounded.…”

Section: Treewidth Versus Clique Numbermentioning

confidence: 99%

“…This gives rise to conceptually simpler proofs of (tw, ω)-boundedness of these two graph classes compared to the proofs from [54,55], which rely on graph minors theory. Furthermore, the fact that the classes of K 2,q -induced-minorfree graphs have bounded tree-independence number (see Section 9) leads to an alternative proof of (tw, ω)-boundedness of these classes of graphs, which also generalizes the fact that the class of graphs with minimal separators of bounded size are (tw, ω)-bounded [107].…”

Section: Related Work On Independent Packingsmentioning

confidence: 99%

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Treewidth versus clique number. II. Tree-independence number

Dallard¹,

Milanič²,

Štorgel³

2021

Preprint

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In 2020, Dallard, Milanič, and Štorgel initiated a systematic study of graph classes in which the treewidth can only be large due the presence of a large clique, which they call (tw, ω)bounded. The family of (tw, ω)-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size. While Chaplick and Zeman showed in 2017 that (tw, ω)-bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, an interesting open problem is whether (tw, ω)-boundedness has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by identifying a sufficient condition for (tw, ω)-bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent H-Packing problem, for any fixed finite set H of connected graphs. This family of problems generalizes several other problems studied in the literature, including the Maximum Weight Independent Set and Maximum Weight Induced Matching problems.Our approach is based on a new min-max graph parameter related to tree decompositions. We define the independence number of a tree decomposition T of a graph as the maximum independence number over all subgraphs of G induced by some bag of T . The tree-independence number of a graph G is then defined as the minimum independence number over all tree decompositions of G. Boundedness of the tree-independence number is a refinement of (tw, ω)-boundedness that is still general enough to hold for all the aforementioned families of graph classes. We show that if a graph is given together with a tree decomposition with bounded independence number, then for any fixed finite set H of connected graphs, the Maximum Weight Independent H-Packing problem can be solved in polynomial time. Motivated by this result, we consider six graph containment relations-the subgraph, topological minor, and minor relations, as well as their induced variants-and for each of them characterize the graphs H for which any graph excluding H with respect to the relation admits a tree decomposition with bounded independence number. These results build on and refine the analogous characterizations due to Dallard, Milanič, and Štorgel for (tw, ω)-boundedness, and shows that in all these cases, (tw, ω)boundedness is actually equivalent to bounded tree-independence number. We use a variety of tools including SPQR trees and potential maximal cliques, and show that in the bounded cases, one can also obtain tree decompositions with bounded independence number efficiently. This leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes, each of which properly contains the class of chordal graphs. These results also apply to the class of...

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“…All these results make it important to identify classes of graphs with a polynomially bounded number of minimal separators. Known classes with this property include chordal graphs [46] and their generalization weakly chordal graphs [11], permutation graphs [6,29] and more generally cocomparability graphs of bounded interval dimension [20], circular-arc graphs [33], circle graphs [30], polygon circle graphs [49], distance-hereditary graphs [31], probe interval graphs [15], AT-free co-AT-free graphs [34], P 4 -sparse graphs [39], extended P 4 -laden graphs [42], and graphs with minimal separators of bounded size [48]. Moreover, it is known that a class of graphs has a polynomially bounded number of minimal separators if and only if it has a polynomially bounded number of potential maximal cliques [12].…”

Section: Introductionmentioning

confidence: 99%

Polynomially Bounding the Number of Minimal Separators in Graphs: Reductions, Sufficient Conditions, and a Dichotomy Theorem

Milanič

Pivač

2021

Electron. J. Combin.

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A graph class is said to be tame if graphs in the class have a polynomially bounded number of minimal separators. Tame graph classes have good algorithmic properties, which follow, for example, from an algorithmic metatheorem of Fomin, Todinca, and Villanger from 2015. We show that a hereditary graph class $\mathcal{G}$ is tame if and only if the subclass consisting of graphs in $\mathcal{G}$ without clique cutsets is tame. This result and Ramsey's theorem lead to several types of sufficient conditions for a graph class to be tame. In particular, we show that any hereditary class of graphs of bounded clique cover number that excludes some complete prism is tame, where a complete prism is the Cartesian product of a complete graph with a $K_2$. We apply these results, combined with constructions of graphs with exponentially many minimal separators, to develop a dichotomy theorem separating tame from non-tame graph classes within the family of graph classes defined by sets of forbidden induced subgraphs with at most four vertices.

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Efficient Analy sis of Graphs with Small Minimal Separators

Cited by 8 publications

References 20 publications

On the maximum weight minimal separator

On the maximum weight minimal separator

Treewidth versus clique number. II. Tree-independence number

Polynomially Bounding the Number of Minimal Separators in Graphs: Reductions, Sufficient Conditions, and a Dichotomy Theorem

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