Threshold Dimension of Graphs

Cozzens, Margaret B.; Leibowitz, Rochelle

doi:10.1137/0605055

Cited by 23 publications

(16 citation statements)

References 14 publications

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“…Similarly, since threshold graphs are closed under complementation, dim T H (G) is the smallest number of threshold graphs using which a graph G can be covered. Chvátal and Hammer introduced the parameter t(G), defined as the smallest number of threshold graphs required to cover a graph G [11], the study of which has resulted in several influential papers [37,13]. The parameter t(G) has been called the "threshold dimension" of G due to the equivalent definition of this parameter as the smallest number of linear inequalities on |V (G)| variables such that every inequality is satisfied by a vector in {0, 1} |V (G)| if and only if it is the characteristic vector of an independent set in G (refer [29] for details).…”

Section: Related Workmentioning

confidence: 99%

Representing Graphs as the Intersection of Cographs and Threshold Graphs

Chacko

Francis

2021

Electron. J. Combin.

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A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\dim_{COG}(G)$ (resp. $\dim_{TH}(G)$) denotes the minimum number of cographs (resp. threshold graphs) whose intersection gives $G$. We present several new bounds on these parameters for general graphs as well as some special classes of graphs. It is shown that for any graph $G$: (a) $\dim_{COG}(G)\leqslant\mathrm{tw}(G)+2$, (b) $\dim_{TH}(G)\leqslant\mathrm{pw}(G)+1$, and (c) $\dim_{TH}(G)\leqslant\chi(G)\cdot\mathrm{box}(G)$, where $\mathrm{tw}(G)$, $\mathrm{pw}(G)$, $\chi(G)$ and $\mathrm{box}(G)$ denote respectively the treewidth, pathwidth, chromatic number and boxicity of the graph $G$. We also derive the exact values for these parameters for cycles and show that every forest is the intersection of two cographs. These results allow us to derive improved bounds on $\dim_{COG}(G)$ and $\dim_{TH}(G)$ when $G$ belongs to some special graph classes.

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Section: Related Workmentioning

confidence: 99%

Representing Graphs as the Intersection of Cographs and Threshold Graphs

Chacko

Francis

2021

Electron. J. Combin.

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“…Finding the minimal clique covering number, or the minimal threshold covering number, is well known to be NP-complete; see e.g. [4].…”

Section: Propositionmentioning

confidence: 99%

Limitations of using tokens for mutual exclusion

Ordman

1995

Proceedings of the 1995 ACM 23rd Annual Conference on Computer Science - CSC '95

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Token-passing algorithms are a common way of solving distributed mutual ezclusion problems in computer networks.We list common properties of mutual exclusion problems and of token-passing algorithms, with a view toward determining what classes of problems can be solved with what sort of tokens. In particular, we delineate when a multiset of tokens, rather than simply a set of tokens, is required. We cite a new example which shows that a mutual ezclusion problem solvable in linear time and space by other means may require exponentially many tokens.

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“…This gave rise to the question of whether there exist any graph G that does not have a threshold cover of size χ(G ′ ). Cozzens and Leibowitz [4] showed the existence of such graphs. In particular, they showed that for every k ≥ 4, there exists a graph G such that χ(G ′ ) = k but G has no threshold cover of size k. The question of whether such graphs exist for k = 2 seems to have been intensely studied but remained open for a decade (see [11]).…”

Section: Introductionmentioning

confidence: 98%

The Lexicographic Method for the Threshold Cover Problem

Francis

Jacob

2020

Algorithms and Discrete Applied Mathematics

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The lexicographic method is a technique that was introduced by Hell and Huang [Journal of Graph Theory, 20(3):361-374, 1995] as a way to simplify the problems of recognizing and obtaining representations of comparability graphs, proper circular-arc graphs and proper interval graphs. This method gives rise to conceptually simple recognition algorithms and leads to much simpler proofs for some characterization theorems for these classes. Threshold graphs are a class of graphs that have many equivalent definitions and have applications in integer programming and set packing problems. A graph is said to have a threshold cover of size k if its edges can be covered using k threshold graphs. Chvátal and Hammer conjectured in 1977 that given a graph G, a suitably constructed auxiliary graph G ′ has chromatic number equal to the minimum size of a threshold cover of G. Although Cozzens and Leibowitz showed that this conjecture is false in the general case, Raschle and Simon [Proceedings of the Twenty-seventh Annual ACM Symposium on Theory of Computing, STOC '95, pages 650-661, 1995] proved that G has a threshold cover of size 2 if and only if G ′ is bipartite. We show how the lexicographic method can be used to obtain a completely new and much simpler proof for this result. This method also gives rise to a simple new LexBFS-based algorithm for recognizing graphs having a threshold cover of size 2. Although this algorithm is not the fastest known, it is a certifying algorithm that matches the time complexity of the fastest known certifying algorithm for this problem. The algorithm can also be easily adapted to give a certifying recognition algorithm for bipartite graphs that can be covered by two chain subgraphs.

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Threshold Dimension of Graphs

Cited by 23 publications

References 14 publications

Representing Graphs as the Intersection of Cographs and Threshold Graphs

Representing Graphs as the Intersection of Cographs and Threshold Graphs

Limitations of using tokens for mutual exclusion

The Lexicographic Method for the Threshold Cover Problem

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