Thinness of product graphs

Bonomo-Braberman, Flavia; Gonzalez, Carolina L.; Oliveira, Fabiano S.; Sampaio, Moysés S.; Szwarcfiter, Jayme L.

doi:10.48550/arxiv.2006.16887

Cited by 2 publications

(3 citation statements)

References 24 publications

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“…These results are presented in Sections 4. 2 and 4.3. This line of research has been previously explored for the following properties of graph products: connectivity [11,12,69,73]; queue-number [70]; stack-number [22,53]; thinness [9]; boxicity and cubicity [13]; polynomial growth [26]; bounded expansion and colouring numbers [26]; chromatic number [20,41,58,61,62,67,68,75]; and Hadwiger number [1,15,37,40,45,49,52,71,74]. Our companion paper [37] studies the Hadwiger number of direct products.…”

Section: Pathwidth and Treewidth Of Direct Productsmentioning

confidence: 99%

Structural Properties of Graph Products

Hickingbotham¹,

Wood²

2021

Preprint

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Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] established that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. Motivated by this result, this paper systematically studies various structural properties of cartesian, direct and strong products. In particular, we characterise when these graph products contain a given complete multipartite subgraph, determine tight bounds for their degeneracy, establish new lower bounds for the treewidth of cartesian and strong products, and characterise when they have bounded treewidth and when they have bounded pathwidth.

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Section: Pathwidth and Treewidth Of Direct Productsmentioning

confidence: 99%

Structural Properties of Graph Products

Hickingbotham¹,

Wood²

2021

Preprint

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“…R 3 ) occurs, which is a contradiction. (v) ⇒ (i)) Since a graph is independent 2-thin if and only if each of its connected components is (see, for example, [8]), we may assume G is connected and non-trivial. Let {V 1 , V 2 } be the bipartition of V (G) and let < be an order of V 1 ∪ V 2 that avoids P 5 , P 6 , and P 9 .…”

Section: Symmetric Patternsmentioning

confidence: 99%

“…Examples of thin representations of graphs are shown in Figure 1. In [8], the concept of (proper) independent thinness was introduced in order to bound the (proper) thinness of the lexicographical and direct products of two graphs. In this case it is required, additionally, the classes of the partition to be independent sets.…”

Section: Introductionmentioning

confidence: 99%

Intersection models and forbidden pattern characterizations for 2-thin and proper 2-thin graphs

Bonomo¹,

Brito²

2021

Preprint

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The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Graphs with thinness at most two include, for example, bipartite convex graphs. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. Proper thinness is defined analogously, generalizing proper interval graphs, and a larger family of NP-complete problems are known to be polynomially solvable for graphs with bounded proper thinness. It is known that the thinness of a graph is at most its pathwidth plus one. In this work, we prove that the proper thinness of a graph is at most its bandwidth, for graphs with at least one edge. It is also known that boxicity is a lower bound for the thinness. The main results of this work are characterizations of 2-thin and 2-proper thin graphs as intersection graphs of rectangles in the plane with sides parallel to the Cartesian axes and other specific conditions. We also bound the bend number of graphs with low thinness as vertex intersection graphs of paths on a grid (B k -VPG graphs are the graphs that have a representation in which each path has at most k bends). We show that 2-thin graphs are a subclass of B 1 -VPG graphs and, moreover, of monotone L-graphs (that is, B 1 -VPG graphs admitting a representation that uses only one of the four possible shapes L, L, L , L , with an extra condition), and that 3-thin graphs are a subclass of B 3 -VPG graphs. We also show that B 0 -VPG graphs may have arbitrarily large thinness, and that not every 4-thin graph is a VPG graph. Finally, we characterize 2-thin graphs by a set of forbidden patterns for a vertex order.

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Thinness of product graphs

Cited by 2 publications

References 24 publications

Structural Properties of Graph Products

Structural Properties of Graph Products

Intersection models and forbidden pattern characterizations for 2-thin and proper 2-thin graphs

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