Boxicity of a graph G(V, E), denoted by box(G), is the minimum integer k such that G can be represented as the intersection graph of axis parallel boxes in R k . The problem of computing boxicity is inapproximable even for graph classes like bipartite, co-bipartite and split graphs within O(n 1−ǫ )-factor, for any ǫ > 0 in polynomial time unless N P = ZP P . We give FPT approximation algorithms for computing the boxicity of graphs, where the parameter used is the vertex or edge edit distance of the given graph from families of graphs of bounded boxicity. This can be seen as a generalization of the parameterizations discussed in [4]. Extending the same idea in one of our algorithms, we also get an O n √ log log n √ log n factor approximation algorithm for computing boxicity and an O n(log log n) 3 2 √ log n factor approximation algorithm for computing the cubicity. These seem to be the first o(n) factor approximation algorithms known for both boxicity and cubicity. As a consequence of this result, a o(n) factor approximation algorithm for computing the partial order dimension of finite posets and a o(n) factor approximation algorithm for computing the threshold dimension of split graphs would follow.
Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). This settles an open problem raised by Biedl [1], in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two dimensional grid. In conjunction with the result of this paper, the constant factor approximation algorithm for this problem obtained by Biedl [1] for 2-vertex-connected outerplanar graphs will work for all outer planar graphs.
Given a point set P and a class C of geometric objects, G C (P) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C ∈ C containing both p and q but no other points from P. We study G ▽ (P) graphs where ▽ is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-Θ 6 graphs and TD-Delaunay graphs.The main result in our paper is that for point sets P in general position, G ▽ (P) always contains a matching of size at least |P|−1 3 and this bound is tight. We also give some structural properties of G (P) graphs, where is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of G (P) is simply a path. Through the equivalence of G (P) graphs with Θ 6 graphs, we also derive that any Θ 6 graph can have at most 5n − 11 edges, for point sets in general position.
Given a point set P and a class C of geometric objects, G C (P) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C ∈ C containing both p and q but no other points from P. We study G ▽ (P) graphs where ▽ is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-Θ 6 graphs and TD-Delaunay graphs.The main result in our paper is that for point sets P in general position, G ▽ (P) always contains a matching of size at leastand this bound is tight. We also give some structural properties of G (P) graphs, where is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of G (P) is simply a path. Through the equivalence of G (P) graphs with Θ 6 graphs, we also derive that any Θ 6 graph can have at most 5n − 11 edges, for point sets in general position.
The eternal vertex cover problem is a variant of the classical vertex cover problem defined in terms of an infinite attacker-defender game played on a graph. In each round of the game, the defender reconfigures guards from one vertex cover to another in response to a move by the attacker. The minimum number of guards required in any winning strategy of the defender when this game is played on a graph G is the eternal vertex cover number of G, denoted by evc(G). It is known that given a graph G and an integer k, checking whether evc(G) ≤ k is NP-hard. Further, it is known that for any graph G, mvc(G) ≤ evc(G) ≤ 2 mvc(G), where mvc(G) is the vertex cover number of G. Though a characterization is known for graphs for which evc(G) = 2 mvc(G), a characterization of graphs for which evc(G) = mvc(G) remained as an open problem, since 2009. We achieve such a characterization for a class of graphs that includes chordal graphs and internally triangulated planar graphs. For biconnected chordal graphs, our characterization leads to a polynomial time algorithm for precisely determining evc(G) and an algorithm for determining a safe strategy for guard movement in each round of the game using only evc(G) guards. Though the eternal vertex cover problem is only known to be in PSPACE in general, it follows from our new characterization that the problem is in NP for locally connected graphs, a graph class which includes all biconnected internally triangulated planar graphs. We also provide reductions establishing NP-completeness of the problem for biconnected internally triangulated planar graphs. As far as we know, this is the first NP-completeness result known for the problem for any graph class.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.