278 research outputs found

    Planar Induced Subgraphs of Sparse Graphs

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    We show that every graph has an induced pseudoforest of at least nm/4.5n-m/4.5 vertices, an induced partial 2-tree of at least nm/5n-m/5 vertices, and an induced planar subgraph of at least nm/5.2174n-m/5.2174 vertices. These results are constructive, implying linear-time algorithms to find the respective induced subgraphs. We also show that the size of the largest KhK_h-minor-free graph in a given graph can sometimes be at most nm/6+o(m)n-m/6+o(m).Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph Algorithms and Application

    Labeling Schemes for Bounded Degree Graphs

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    We investigate adjacency labeling schemes for graphs of bounded degree Δ=O(1)\Delta = O(1). In particular, we present an optimal (up to an additive constant) logn+O(1)\log n + O(1) adjacency labeling scheme for bounded degree trees. The latter scheme is derived from a labeling scheme for bounded degree outerplanar graphs. Our results complement a similar bound recently obtained for bounded depth trees [Fraigniaud and Korman, SODA 10], and may provide new insights for closing the long standing gap for adjacency in trees [Alstrup and Rauhe, FOCS 02]. We also provide improved labeling schemes for bounded degree planar graphs. Finally, we use combinatorial number systems and present an improved adjacency labeling schemes for graphs of bounded degree Δ\Delta with (e+1)n<Δn/5(e+1)\sqrt{n} < \Delta \leq n/5

    A Variant of the Maximum Weight Independent Set Problem

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    We study a natural extension of the Maximum Weight Independent Set Problem (MWIS), one of the most studied optimization problems in Graph algorithms. We are given a graph G=(V,E)G=(V,E), a weight function w:VR+w: V \rightarrow \mathbb{R^+}, a budget function b:VZ+b: V \rightarrow \mathbb{Z^+}, and a positive integer BB. The weight (resp. budget) of a subset of vertices is the sum of weights (resp. budgets) of the vertices in the subset. A kk-budgeted independent set in GG is a subset of vertices, such that no pair of vertices in that subset are adjacent, and the budget of the subset is at most kk. The goal is to find a BB-budgeted independent set in GG such that its weight is maximum among all the BB-budgeted independent sets in GG. We refer to this problem as MWBIS. Being a generalization of MWIS, MWBIS also has several applications in Scheduling, Wireless networks and so on. Due to the hardness results implied from MWIS, we study the MWBIS problem in several special classes of graphs. We design exact algorithms for trees, forests, cycle graphs, and interval graphs. In unweighted case we design an approximation algorithm for d+1d+1-claw free graphs whose approximation ratio (dd) is competitive with the approximation ratio (d2\frac{d}{2}) of MWIS (unweighted). Furthermore, we extend Baker's technique \cite{Baker83} to get a PTAS for MWBIS in planar graphs.Comment: 18 page

    Graph Treewidth and Geometric Thickness Parameters

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    Consider a drawing of a graph GG in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of GG, is the classical graph parameter "thickness". By restricting the edges to be straight, we obtain the "geometric thickness". By further restricting the vertices to be in convex position, we obtain the "book thickness". This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth kk, the maximum thickness and the maximum geometric thickness both equal k/2\lceil{k/2}\rceil. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth kk, the maximum book thickness equals kk if k2k \leq 2 and equals k+1k+1 if k3k \geq 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in Computer Science 3843:129-140, Springer, 2006. The full version was published in Discrete & Computational Geometry 37(4):641-670, 2007. That version contained a false conjecture, which is corrected on page 26 of this versio

    Graph classes and forbidden patterns on three vertices

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    This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.Comment: Third version version. 38 page

    The edge chromatic number of outer-1-planar graphs

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    A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. In this paper, we completely determine the edge chromatic number of outer 1-planar graphs

    Vertex-Coloring with Star-Defects

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    Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, and acyclicity of the monochromatic components. In this paper we focus on defective colorings in which the monochromatic components are acyclic and have small diameter, namely, they form stars. For outerplanar graphs, we give a linear-time algorithm to decide if such a defective coloring exists with two colors and, in the positive case, to construct one. Also, we prove that an outerpath (i.e., an outerplanar graph whose weak-dual is a path) always admits such a two-coloring. Finally, we present NP-completeness results for non-planar and planar graphs of bounded degree for the cases of two and three colors
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