Packing edge disjoint cliques in graphs

The classical disjoint shortest path problem has recently recalled interests from researchers in the network planning and optimization community. However, the requirement of the shortest paths being completely vertex or edge disjoint might be too restrictive and demands much more resources in a network. Partially disjoint shortest paths, in which a bounded number of shared vertices or edges is allowed, balance between degree of disjointness and occupied network resources. In this paper, we consider the problem of finding k shortest paths which are edge disjoint but partially vertex disjoint. For a pair of distinct vertices in a network graph, the problem aims to optimally find k edge disjoint shortest paths among which at most a bounded number of vertices are shared by at least two paths. In particular, we present novel techniques for exactly solving the problem with a runtime that significantly improves the current best result. The proposed algorithm is also validated by computer experiments on both synthetic and real networks which demonstrate its superior efficiency of up to three orders of magnitude faster than the state of the art.

A set A of vertices of an undirected graph G is called k‐edge‐connected in G if for all pairs of distinct vertices a, b∈A, there exist k edge disjoint a, b‐paths in G. An A‐tree is a subtree of G containing A, and an A‐bridge is a subgraph B of G which is either formed by a single edge with both end vertices in A or formed by the set of edges incident with the vertices of some component of G − A. It is proved that (i) if A is k·(ℓ + 2)‐edge‐connected in G and every A‐bridge has at most ℓ vertices in V(G) − A or at most ℓ + 2 vertices in A then there exist k edge disjoint A‐trees, and that (ii) if A is k‐edge‐connected in G and B is an A‐bridge such that B is a tree and every vertex in V(B) − A has degree 3 then either A is k‐edge‐connected in G − e for some e∈E(B) or A is (k − 1)‐edge‐connected in G − E(B). © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 188–198, 2009

The disjoint shortest paths problem

We study four NP-hard optimal seat arrangement problems which each have as input a set of n agents, where each agent has cardinal preferences over other agents, and an n-vertex undirected graph (called the seat graph). The task is to assign each agent to a distinct vertex in the seat graph such that either the sum of utilities or the minimum utility is maximized, or it is envy-free or exchange-stable. Aiming at identifying hard and easy cases, we extensively study the algorithmic complexity of the four problems by looking into natural graph classes for the seat graph (e.g., paths, cycles, stars, or matchings), problem-specific parameters (e.g., the number of non-isolated vertices in the seat graph or the maximum number of agents towards whom an agent has non-zero preferences), and preference structures (e.g., non-negative or symmetric preferences). For strict preferences and seat graphs with disjoint edges and isolated vertices, we correct an error in the literature and show that finding an envy-free arrangement remains NP-hard in this case.

A non-perfect maze is a maze that contains loop or cycle and has no isolated cell. A non-perfect maze is an alternative to obtain a maze that cannot be satisfied by perfect maze. This paper discusses non-perfect maze generation with two kind of biases, that is, horizontal and vertical wall bias and cycle bias. In this research, a maze is modeled as a graph in order to generate non-perfect maze using Kruskal algorithm modifications. The modified Kruskal algorithm used Fisher Yates algorithm to obtain a random edge sequence and disjoint set data structure to reduce process time of the algorithm. The modification mentioned above are adding edges randomly while taking account of the edge’s orientation, and by adding additional edges after spanning tree is formed. The algorithm designed in this research constructs an non-perfect maze with complexity of where and denote vertex and edge set of an grid graph, respectively. Several biased non-perfect mazes were shown in this research by varying its dimension, wall bias and cycle bias.

The main objective of this research article is to classify different types of m-polar fuzzy edges in an m-polar fuzzy graph by using the strength of connectedness between pairs of vertices. The identification of types of m-polar fuzzy edges, including α-strong m-polar fuzzy edges, β-strong m-polar fuzzy edges and δ-weak m-polar fuzzy edges proved to be very useful to completely determine the basic structure of m-polar fuzzy graph. We analyze types of m-polar fuzzy edges in strongest m-polar fuzzy path and m-polar fuzzy cycle. Further, we define various terms, including m-polar fuzzy cut-vertex, m-polar fuzzy bridge, strength reducing set of vertices and strength reducing set of edges. We highlight the difference between edge disjoint m-polar fuzzy path and internally disjoint m-polar fuzzy path from one vertex to another vertex in an m-polar fuzzy graph. We define strong size of an m-polar fuzzy graph. We then present the most celebrated result due to Karl Menger for m-polar fuzzy graphs and illustrate the vertex version of Menger’s theorem to find out the strongest m-polar fuzzy paths between affected and non-affected cities of a country due to an earthquake. Moreover, we discuss an application of types of m-polar fuzzy edges to determine traffic-accidental zones in a road network. Finally, a comparative analysis of our research work with existing techniques is presented to prove its applicability and effectiveness.

A matching theorem for graphs

Variations of coloring problems related to scheduling and discrete tomography

We investigate the existence of edge-magic labellings of countably infinite graphs by abelian groups. We show for that for a large class of abelian groups, including the integers ${\Bbb Z}$, there is such a labelling whenever the graph has an infinite set of disjoint edges. A graph without an infinite set of disjoint edges must be some subgraph of $H + {\cal I}$, where $H$ is some finite graph and ${\cal I}$ is a countable set of isolated vertices. Using power series of rational functions, we show that any edge-magic ${\Bbb Z}$-labelling of $H + {\cal I}$ has almost all vertex labels making up pairs of half-modulus classes. We also classify all possible edge-magic ${\Bbb Z}$-labellings of $H + {\cal I}$ under the assumption that the vertices of the finite graph are labelled consecutively.

This thesis is devoted to the understanding of topological graphs. We consider the following four problems: 1. A simple topological graph is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once, at endpoint or at a crossing. Let $G$ be a complete simple topological graph on $n$ vertices. The three edges induced by any triplet of vertices in $G$ form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an empty triangle. In 1998, Harborth proved that $G$ has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least $2n/3$. We settle Harborth's conjecture in the affirmative. 2. A monotone cylindrical graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called simple if any pair of its edges have at most one point in common: an endpoint or a point at which they properly cross. We say that two edges are disjoint if they do not intersect. We show that every simple complete monotone cylindrical graph on $n$ vertices contains $\Omega(n^{1-\epsilon})$ pairwise disjoint edges for any $\epsilon>0$. As a consequence, we show that every simple complete topological graph (drawn in the plane) with $n$ vertices contains $\Omega(n^{\frac 12-\epsilon})$ pairwise disjoint edges for any $\epsilon>0$. By extending some of the ideas here we are then able to get rid of the $\epsilon$ term in the exponent, showing that in fact we can always guarantee a set with $\Omega(n^{\frac 12})$ pairwise disjoint edges. This improves the previous lower bound of $\Omega(n^\frac 13)$ by Suk and independently by Fulek. We remark that our proof implies a polynomial time algorithm for finding this set of pairwise disjoint edges. 3. A {\em tangle} is a graph drawn in the plane such that its edges are represented by continuous arcs, and any two edges share precisely one point, which is either a common endpoint or an interior point at which the two edges are tangent to each other. These points of tangencies are assumed to be distinct. If we drop the last assumption, that is, more than two edges may touch one another at the same point, then the drawing is called a {\em degenerate tangle}. We settle a problem of Pach, Radoi\v ci\'c, and T\'oth \cite{TTpaper}, by showing that every degenerate tangle has at most as many edges as vertices. We also give a complete characterization of tangles. 4. We show that for a constant $t\in \NN$, every simple topological graph on $n$ vertices has $O(n)$ edges if the graph has no two sets of $t$ edges each such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is $K_{t,t}$-free). As an application, we settle the tangled-thrackle conjecture formulated by Pach, Radoi\v{c}i\'c, and T\'oth: Every $n$-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most $O(n)$ edges.