Computing the boxicity of a graph by covering its complement by cointerval graphs

Cozzens, Margaret B.; Roberts, Fred S.

doi:10.1016/0166-218x(83)90077-x

Cited by 59 publications

(67 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Boxicity has been investigated for various classes of graphs [5][13] [14] and has been related with other parameters such as treewidth [15] and vertex cover [16]. Computing the boxicity of a graph was shown to be NP-hard by Cozzens [9]. This was later strengthened by Yannakakis [6], and finally by Kratochvil [8] who showed that deciding whether boxicity of a graph is at most two itself is NP-complete.…”

Section: Introductionmentioning

confidence: 87%

Boxicity of Circular Arc Graphs

Bhowmick

Chandran

2010

Graphs and Combinatorics

View full text Add to dashboard Cite

A k-dimensional box is the cartesian product R 1 × R 2 × · · · × R k where each R i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes: that is two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. Let G be a circular arc graph with maximum degree ∆. We show that if ∆ α but with ∆ = n (α−1) 2α + n 2α(α+1) + (α + 2). So the result cannot be improved substantially when α is large. Let r inf be minimum number of arcs passing through any point on the circle with respect to some circular arc representation of G. We also show that for any circular arc graph G, box(G) ≤ r inf + 1 and this bound is tight. Given a family of arcs F on the circle, the circular cover number L(F ) is the cardinality of the smallest subset F ′ of F such that the arcs in F ′ can cover the circle. Maximum circular cover number L max (G) is defined as the maximum value of L(F ) obtained over all possible family of arcs F that can represent G. We will show that if G is a circular arc graph with L max (G) > 4 then box(G) ≤ 3.

show abstract

Section: Introductionmentioning

confidence: 87%

Boxicity of Circular Arc Graphs

Bhowmick

Chandran

2010

Graphs and Combinatorics

View full text Add to dashboard Cite

show abstract

“…The term boxicity is well-defined [5], but hard to compute [5]. There are fast algorithms to test if a graph is an interval graph, but if it is not an interval graph then there are no fast ways of telling the boxicity of the graph.…”

Section: Interval Graphs and Habitat Dimensionmentioning

confidence: 99%

“…It would be easy to do so, for example, consider the possible correspondence: Figure 6: G and a demonstration that G is an interval graph. [1,4] e to [1.5, 6] f to [5,7] g to [9,11] Remark 9. Note The size of the interval makes no difference, nor does whether the intervals are open (do not include the endpoints) or are closed (include the endpoints).…”

Section: Interval Graphs and Habitat Dimensionmentioning

confidence: 99%

Food Webs, Competition Graphs, and Habitat Formation

Cozzens

2011

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

Abstract. One interesting example of a discrete mathematical model used in biology is a food web. The first biology courses in high school and in college present the fundamental nature of a food web, one that is understandable by students at all levels. But food webs as part of a larger system are often not addressed. This paper presents materials that can be used in undergraduate classes in biology (and mathematics) and provides students with the opportunity to explore mathematical models of predator-prey relationships, determine trophic levels, dominant species, stability of the ecosystem, competition graphs, interval graphs, and even confront problems that would appear to have logical answers that are as yet unsolved.Key words: food web, predators and prey, dominance, trophic level, trophic status, mathematical model, graph, directed graph, competition graph, interval graph, boxicity AMS subject classification: Goals:• Recognize various relationships between organisms, and look for patterns in food webs.• Use graphs and directed graphs to model complex trophic relationships.• Determine trophic levels and status within a food web, and the significance of these levels in calculating the relative importance of each species (vertices) and each relationship (arcs) in a food web.• Use the food web and corresponding competition graph to determine the dimension of the community's habitat.

show abstract

“…Graphs with box(G) ≤ 1 are interval graphs, which can be easily identified in linear time [Booth and Lueker 1976;Habib et al 2000]. Algorithms exist to test if box(G) ≤ 2 [Quest and Wegner 1990] or to compute boxicity in general [Cozzens and Roberts 1983], but they are a lot more cumbersome. The task of testing if box(G) ≤ d is known to be NP-complete for all d ≥ 2 [Cozzens and Roberts 1983].…”

Section: Boxes and Agreeable Graphsmentioning

confidence: 99%

Untitled

2010

Involve

View full text Add to dashboard Cite

Computing the boxicity of a graph by covering its complement by cointerval graphs

Cited by 59 publications

References 6 publications

Boxicity of Circular Arc Graphs

Boxicity of Circular Arc Graphs

Food Webs, Competition Graphs, and Habitat Formation

Untitled

Contact Info

Product

Resources

About