42 research outputs found

    Local Boxicity, Local Dimension, and Maximum Degree

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    In this paper, we focus on two recently introduced parameters in the literature, namely `local boxicity' (a parameter on graphs) and `local dimension' (a parameter on partially ordered sets). We give an `almost linear' upper bound for both the parameters in terms of the maximum degree of a graph (for local dimension we consider the comparability graph of a poset). Further, we give an O(nΔ2)O(n\Delta^2) time deterministic algorithm to compute a local box representation of dimension at most 3Δ3\Delta for a claw-free graph, where nn and Δ\Delta denote the number of vertices and the maximum degree, respectively, of the graph under consideration. We also prove two other upper bounds for the local boxicity of a graph, one in terms of the number of vertices and the other in terms of the number of edges. Finally, we show that the local boxicity of a graph is upper bounded by its `product dimension'.Comment: 11 page

    Box representations of embedded graphs

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    A dd-box is the cartesian product of dd intervals of R\mathbb{R} and a dd-box representation of a graph GG is a representation of GG as the intersection graph of a set of dd-boxes in Rd\mathbb{R}^d. It was proved by Thomassen in 1986 that every planar graph has a 3-box representation. In this paper we prove that every graph embedded in a fixed orientable surface, without short non-contractible cycles, has a 5-box representation. This directly implies that there is a function ff, such that in every graph of genus gg, a set of at most f(g)f(g) vertices can be removed so that the resulting graph has a 5-box representation. We show that such a function ff can be made linear in gg. Finally, we prove that for any proper minor-closed class F\mathcal{F}, there is a constant c(F)c(\mathcal{F}) such that every graph of F\mathcal{F} without cycles of length less than c(F)c(\mathcal{F}) has a 3-box representation, which is best possible.Comment: 16 pages, 6 figures - revised versio

    Structural parameterizations for boxicity

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    The boxicity of a graph GG is the least integer dd such that GG has an intersection model of axis-aligned dd-dimensional boxes. Boxicity, the problem of deciding whether a given graph GG has boxicity at most dd, is NP-complete for every fixed d2d \ge 2. We show that boxicity is fixed-parameter tractable when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Adiga et al., that boxicity is fixed-parameter tractable in the vertex cover number. Moreover, we show that boxicity admits an additive 11-approximation when parameterized by the pathwidth of the input graph. Finally, we provide evidence in favor of a conjecture of Adiga et al. that boxicity remains NP-complete when parameterized by the treewidth.Comment: 19 page

    Revisiting Interval Graphs for Network Science

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    The vertices of an interval graph represent intervals over a real line where overlapping intervals denote that their corresponding vertices are adjacent. This implies that the vertices are measurable by a metric and there exists a linear structure in the system. The generalization is an embedding of a graph onto a multi-dimensional Euclidean space and it was used by scientists to study the multi-relational complexity of ecology. However the research went out of fashion in the 1980s and was not revisited when Network Science recently expressed interests with multi-relational networks known as multiplexes. This paper studies interval graphs from the perspective of Network Science

    Separation dimension of bounded degree graphs

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    The 'separation dimension' of a graph GG is the smallest natural number kk for which the vertices of GG can be embedded in Rk\mathbb{R}^k such that any pair of disjoint edges in GG can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family F\mathcal{F} of total orders of the vertices of GG such that for any two disjoint edges of GG, there exists at least one total order in F\mathcal{F} in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on nn vertices is Θ(logn)\Theta(\log n). In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree dd is at most 29logdd2^{9log^{\star} d} d. We also demonstrate that the above bound is nearly tight by showing that, for every dd, almost all dd-regular graphs have separation dimension at least d/2\lceil d/2\rceil.Comment: One result proved in this paper is also present in arXiv:1212.675

    On local structures of cubicity 2 graphs

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    A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the XX-axis, distance 1+ϵ1 + \epsilon (0<ϵ<10 < \epsilon < 1) apart. This family of graphs allow us to study local structures of unit square intersection graphs, that is, graphs with cubicity 2. The complexity of determining whether a tree has cubicity 2 is unknown while the graph recognition problem for unit square intersection graph is known to be NP-hard. We present a polynomial time algorithm for recognizing trees that admit a 2SUIG representation

    Matchings, coverings, and Castelnuovo-Mumford regularity

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    We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.Comment: 12 pages; v4 has minor changes for publicatio
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