Planar subgraphs without low-degree nodes

We study the following problem: given a geometric graph G and an integer k, determine if G has a planar spanning subgraph (with the original embedding and straight-line edges) such that all nodes have degree at least k. If G is a unit disk graph, the problem is trivial to solve for k = 1. We show that even the slightest deviation from the trivial case (e.g., quasi unit disk graphs or k = 1) leads to NP-hard problems.Peer reviewe

Crossing Patterns in Nonplanar Road Networks

We define the crossing graph of a given embedded graph (such as a road network) to be a graph with a vertex for each edge of the embedding, with two crossing graph vertices adjacent when the corresponding two edges of the embedding cross each other. In this paper, we study the sparsity properties of crossing graphs of real-world road networks. We show that, in large road networks (the Urban Road Network Dataset), the crossing graphs have connected components that are primarily trees, and that the remaining non-tree components are typically sparse (technically, that they have bounded degeneracy). We prove theoretically that when an embedded graph has a sparse crossing graph, it has other desirable properties that lead to fast algorithms for shortest paths and other algorithms important in geographic information systems. Notably, these graphs have polynomial expansion, meaning that they and all their subgraphs have small separators.Comment: 9 pages, 4 figures. To appear at the 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems(ACM SIGSPATIAL 2017

Orthogonal Graph Drawing with Inflexible Edges

We consider the problem of creating plane orthogonal drawings of 4-planar graphs (planar graphs with maximum degree 4) with constraints on the number of bends per edge. More precisely, we have a flexibility function assigning to each edge $e$ a natural number $\mathrm{flex}(e)$, its flexibility. The problem FlexDraw asks whether there exists an orthogonal drawing such that each edge $e$ has at most $\mathrm{flex}(e)$ bends. It is known that FlexDraw is NP-hard if $\mathrm{flex}(e) = 0$ for every edge $e$. On the other hand, FlexDraw can be solved efficiently if $\mathrm{flex}(e) \ge 1$ and is trivial if $\mathrm{flex}(e) \ge 2$ for every edge $e$. To close the gap between the NP-hardness for $\mathrm{flex}(e) = 0$ and the efficient algorithm for $\mathrm{flex}(e) \ge 1$, we investigate the computational complexity of FlexDraw in case only few edges are inflexible (i.e., have flexibility~$0$). We show that for any $\varepsilon > 0$ FlexDraw is NP-complete for instances with $O(n^\varepsilon)$ inflexible edges with pairwise distance $\Omega(n^{1-\varepsilon})$ (including the case where they induce a matching). On the other hand, we give an FPT-algorithm with running time $O(2^k\cdot n \cdot T_{\mathrm{flow}}(n))$, where $T_{\mathrm{flow}}(n)$ is the time necessary to compute a maximum flow in a planar flow network with multiple sources and sinks, and $k$ is the number of inflexible edges having at least one endpoint of degree 4.Comment: 23 pages, 5 figure

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus, and Minor-Free Graphs

A (1 + eps)-approximate distance oracle for a graph is a data structure that supports approximate point-to-point shortest-path-distance queries. The most relevant measures for a distance-oracle construction are: space, query time, and preprocessing time. There are strong distance-oracle constructions known for planar graphs (Thorup, JACM'04) and, subsequently, minor-excluded graphs (Abraham and Gavoille, PODC'06). However, these require Omega(eps^{-1} n lg n) space for n-node graphs. We argue that a very low space requirement is essential. Since modern computer architectures involve hierarchical memory (caches, primary memory, secondary memory), a high memory requirement in effect may greatly increase the actual running time. Moreover, we would like data structures that can be deployed on small mobile devices, such as handhelds, which have relatively small primary memory. In this paper, for planar graphs, bounded-genus graphs, and minor-excluded graphs we give distance-oracle constructions that require only O(n) space. The big O hides only a fixed constant, independent of \epsilon and independent of genus or size of an excluded minor. The preprocessing times for our distance oracle are also faster than those for the previously known constructions. For planar graphs, the preprocessing time is O(n lg^2 n). However, our constructions have slower query times. For planar graphs, the query time is O(eps^{-2} lg^2 n). For our linear-space results, we can in fact ensure, for any delta > 0, that the space required is only 1 + delta times the space required just to represent the graph itself

Graphs with Plane Outside-Obstacle Representations

An \emph{obstacle representation} of a graph consists of a set of polygonal obstacles and a distinct point for each vertex such that two points see each other if and only if the corresponding vertices are adjacent. Obstacle representations are a recent generalization of classical polygon--vertex visibility graphs, for which the characterization and recognition problems are long-standing open questions. In this paper, we study \emph{plane outside-obstacle representations}, where all obstacles lie in the unbounded face of the representation and no two visibility segments cross. We give a combinatorial characterization of the biconnected graphs that admit such a representation. Based on this characterization, we present a simple linear-time recognition algorithm for these graphs. As a side result, we show that the plane vertex--polygon visibility graphs are exactly the maximal outerplanar graphs and that every chordal outerplanar graph has an outside-obstacle representation.Comment: 12 pages, 7 figure

Extended core and choosability of a graph

A graph $G$ is $(a,b)$-choosable if for any color list of size $a$ associated with each vertices, one can choose a subset of $b$ colors such that adjacent vertices are colored with disjoint color sets. This paper shows an equivalence between the $(a,b)$-choosability of a graph and the $(a,b)$-choosability of one of its subgraphs called the extended core. As an application, this result allows to prove the $(5,2)$-choosability and $(7,3)$-colorability of triangle-free induced subgraphs of the triangular lattice.Comment: 10 page

Coalition structure generation over graphs

We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) → R over the subsets of nodes, the problem is to find a partition of N into connected subsets, that maximises the sum of the components values. This problem is generally NP-complete; in particular, it is hard for a defined class of valuation functions which are independent of disconnected members — that is, two nodes have no effect on each others marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive linear time bounds for graphs of bounded treewidth. However, as we show, the problem remains NP-complete for planar graphs, and hence, for any Kk minor free graphs where k ≥ 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m2) nodes. Importantly, our hardness result holds for a particular subclass of valuation functions, termed edge sum, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph