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

Distributed Detection of Cycles

Distributed property testing in networks has been introduced by Brakerski and Patt-Shamir (2011), with the objective of detecting the presence of large dense sub-networks in a distributed manner. Recently, Censor-Hillel et al. (2016) have shown how to detect 3-cycles in a constant number of rounds by a distributed algorithm. In a follow up work, Fraigniaud et al. (2016) have shown how to detect 4-cycles in a constant number of rounds as well. However, the techniques in these latter works were shown not to generalize to larger cycles $C_k$ with $k\geq 5$. In this paper, we completely settle the problem of cycle detection, by establishing the following result. For every $k\geq 3$, there exists a distributed property testing algorithm for $C_k$-freeness, performing in a constant number of rounds. All these results hold in the classical CONGEST model for distributed network computing. Our algorithm is 1-sided error. Its round-complexity is $O(1/\epsilon)$ where $\epsilon\in(0,1)$ is the property testing parameter measuring the gap between legal and illegal instances

Forbidden Directed Minors and Kelly-width

Partial 1-trees are undirected graphs of treewidth at most one. Similarly, partial 1-DAGs are directed graphs of KellyWidth at most two. It is well-known that an undirected graph is a partial 1-tree if and only if it has no K_3 minor. In this paper, we generalize this characterization to partial 1-DAGs. We show that partial 1-DAGs are characterized by three forbidden directed minors, K_3, N_4 and M_5

On a class of intersection graphs

Given a directed graph D = (V,A) we define its intersection graph I(D) = (A,E) to be the graph having A as a node-set and two nodes of I(D) are adjacent if their corresponding arcs share a common node that is the tail of at least one of these arcs. We call these graphs facility location graphs since they arise from the classical uncapacitated facility location problem. In this paper we show that facility location graphs are hard to recognize and they are easy to recognize when the graph is triangle-free. We also determine the complexity of the vertex coloring, the stable set and the facility location problems on that class

Distributed Testing of Excluded Subgraphs

We study property testing in the context of distributed computing, under the classical CONGEST model. It is known that testing whether a graph is triangle-free can be done in a constant number of rounds, where the constant depends on how far the input graph is from being triangle-free. We show that, for every connected 4-node graph H, testing whether a graph is H-free can be done in a constant number of rounds too. The constant also depends on how far the input graph is from being H-free, and the dependence is identical to the one in the case of testing triangles. Hence, in particular, testing whether a graph is K_4-free, and testing whether a graph is C_4-free can be done in a constant number of rounds (where K_k denotes the k-node clique, and C_k denotes the k-node cycle). On the other hand, we show that testing K_k-freeness and C_k-freeness for k>4 appear to be much harder. Specifically, we investigate two natural types of generic algorithms for testing H-freeness, called DFS tester and BFS tester. The latter captures the previously known algorithm to test the presence of triangles, while the former captures our generic algorithm to test the presence of a 4-node graph pattern H. We prove that both DFS and BFS testers fail to test K_k-freeness and C_k-freeness in a constant number of rounds for k>4

Implementing path coloring algorithms on planar graphs

Master's Project (M.S.) University of Alaska Fairbanks, 2017A path coloring of a graph partitions its vertex set into color classes such that each class induces a disjoint union of paths. In this project we implement several algorithms to compute path colorings of graphs embedded in the plane. We present two algorithms to path color plane graphs with 3 colors based on a proof by Poh in 1990. First we describe a naive algorithm that directly follows Poh's procedure, then we give a modified algorithm that runs in linear time. Independent results of Hartman and Skrekovski describe a procedure that takes a plane graph G and a list of 3 colors for each vertex, and computes a path coloring of G such that each vertex receives a color from its list. We present a linear time implementation based on Hartman and Skrekovski's proofs. A C++ implementation is provided for all three algorithms, utilizing the Boost Graph Library. Instructions are given on how to use the implementation to construct colorings for plane graphs represented by Boost data structures