Showing papers on "Path graph published in 2015"

Spectral radius and Hamiltonian properties of graphs

Abstract: Let be a graph with minimum degree . The spectral radius of , denoted by , is the largest eigenvalue of the adjacency matrix of . In this note, we mainly prove the following two results.(1) Let be a graph on vertices with . If , then contains a Hamilton path unless .(2) Let be a graph on vertices with . If , then contains a Hamilton cycle unless . As corollaries of our first result, two previous theorems due to Fiedler and Nikiforov and Lu et al. are obtained, respectively. Our second result refines another previous theorem of Fiedler and Nikiforov.

On the dot product graph of a commutative ring

Abstract: Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × … ×A (n times). The total dot product graph of R is the (undirected) graph TD(R) with vertices R* = R∖{(0, 0,…, 0)}, and two distinct vertices x and y are adjacent if and only if x·y = 0 ∈ A (where x·y denote the normal dot product of x and y). Let Z(R) denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of TD(R) with vertices Z(R)* = Z(R)∖{(0, 0,…, 0)}. It follows that each edge (path) of the classical zero-divisor graph Γ(R) is an edge (path) of ZD(R). We observe that if n = 1, then TD(R) is a disconnected graph and ZD(R) is identical to the well-known zero-divisor graph of R in the sense of Beck–Anderson–Livingston, and hence it is connected. In this paper, we study both graphs TD(R) and ZD(R). For a commutative ring A and n ≥ 3, we show that TD(R) (ZD(R)) is connected with diameter two (at most three) and with girth three. Among other things, for ...

Adjacency Labeling Schemes and Induced-Universal Graphs

Abstract: We describe a way of assigning labels to the vertices of any undirected graph on up to n vertices, each composed of n/2+O(1) bits, such that given the labels of two vertices, and no other information regarding the graph, it is possible to decide whether or not the vertices are adjacent in the graph. This is optimal, up to an additive constant, and constitutes the first improvement in almost 50 years of an n/2+O(log n) bound of Moon. As a consequence, we obtain an induced-universal graph for n-vertex graphs containing only O(2n/2) vertices, which is optimal up to a multiplicative constant, solving an open problem of Vizing from 1968. We obtain similar tight results for directed graphs, tournaments and bipartite graphs.

New bounds on the maximum number of edges in k-quasi-planar graphs

Abstract: A topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed k, the maximum number of edges in a k-quasi-planar graph on n vertices is O ( n ) . Fox and Pach showed that every k-quasi-planar graph with n vertices has at most n ( log ? n ) O ( log ? k ) edges. We improve this upper bound to 2 α ( n ) c n log ? n , where α ( n ) denotes the inverse Ackermann function and c depends only on k, for k-quasi-planar graphs in which any two edges intersect in a bounded number of points. We also show that every k-quasi-planar graph with n vertices in which any two edges have at most one point in common has at most O ( n log ? n ) edges. This improves the previously known upper bound of 2 α ( n ) c n log ? n obtained by Fox, Pach, and Suk.

Long paths and cycles in random subgraphs of graphs with large minimum degree

Abstract: For a given finite graph G of minimum degree at least k, let Gp be a random subgraph of G obtained by taking each edge independently with probability p. We prove that i if pi¾?ω/k for a function ω=ωk that tends to infinity as k does, then Gp asymptotically almost surely contains a cycle and thus a path of length at least 1-o1k, and ii if pi¾?1+o1lnk/k, then Gp asymptotically almost surely contains a path of length at least k. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking G to be the complete graph on k + 1 vertices. © Wiley Periodicals, Inc. Random Struct. Alg., 46, 320-345, 2015

2-edge connectivity in directed graphs

Abstract: Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly not much has been investigated for directed graphs. In this paper we study 2-edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices v and w are 2-edge-connected if there are two edge-disjoint paths from v to w and two edge-disjoint paths from w to v. This relation partitions the vertices into blocks such that all vertices in the same block are 2-edge-connected. Differently from the undirected case, those blocks do not correspond to the 2-edge-connected components of the graph. The main result of this paper is an algorithm for computing the 2-edge-connected blocks of a directed graph in linear time. Besides being asymptotically optimal, our algorithm improves significantly over previous bounds. Once the 2-edge-connected blocks are available, we can test in constant time if two vertices are 2-edge-connected. Additionally, we also show how to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has O(n) edges and maintains the same 2-edge-connected blocks as the input graph, where n is the number of vertices.

Note on the Hardness of Rainbow Connections for Planar and Line Graphs

Abstract: An edge-colored graph $$G$$ is rainbow connected if every two vertices are connected by a path whose edges have distinct colors. It is known that deciding whether a given edge-colored graph is rainbow connected is NP-complete. We will prove that it is still NP-complete even when the edge-colored graph is a planar bipartite graph. A vertex-colored graph is rainbow vertex-connected if every two vertices are connected by a path whose internal vertices have distinct colors. It is known that deciding whether a given vertex-colored graph is rainbow vertex-connected is NP-complete. We will prove that it is still NP-complete even when the vertex-colored graph is a line graph.

On the Anti-Forcing Number of Fullerene Graphs

Abstract: The anti-forcing number of a connected graph G is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching. In this paper, we show that the anti-forcing number of every fullerene has at least four. We give a procedure to construct all fullerenes whose anti-forcing numbers achieve the lower bound four. Furthermore, we show that, for every even n 20 (n6 22; 26), there exists a fullerene with n vertices that has the anti-forcing number four, and the fullerene with 26 vertices has the anti-forcing number ve.

Parameterized Lower Bound and Improved Kernel for Diamond-free Edge Deletion

Abstract: A diamond is a graph obtained by removing an edge from a complete graph on four vertices. A graph is diamond-free if it does not contain an induced diamond. The Diamond-free Edge Deletion problem asks to find whether there exist at most k edges in the input graph whose deletion results in a diamond-free graph. The problem was proved to be NP-complete and a polynomial kernel of O(k^4) vertices was found by Fellows et. al. (Discrete Optimization, 2011). In this paper, we give an improved kernel of O(k^3) vertices for Diamond-free Edge Deletion. We give an alternative proof of the NP-completeness of the problem and observe that it cannot be solved in time 2^{o(k)} * n^{O(1)}, unless the Exponential Time Hypothesis fails.

Graphs with fixed number of pendent vertices and minimal first Zagreb index

Abstract: The rst Zagreb index M1 of a graph G is equal to the sum of squares of degrees of the vertices of G. Goubko proved that for trees with n1 pendent vertices, M1 9n1 16. We show how this result can be extended to hold for any connected graph with cyclomatic number 0. In addition, graphs with n vertices, n1 pendent vertices, cyclomatic number , and minimal M1 are characterized. Explicit expressions for minimal M1 are given for = 0;1;2, which directly can be extended for > 2.

The Turan Number of Disjoint Copies of Paths

Abstract: The Tur\'{a}n number of a graph $H$, $ex(n,H)$, is the maximum number of edges in a simple graph of order $n$ which does not contain $H$ as a subgraph. Let $k\cdot P_3$ denote $k$ disjoint copies of a path on $3$ vertices. In this paper, we determine the value $ex(n, k\cdot P_3)$ and characterize all extremal graphs. This extends a result of Bushaw and Kettle [N. Bushaw and N. Kettle, Tur\'{a}n Numbers of multiple and equibipartite forests, Combin. Probab. Comput., 20(2011) 837-853.], which solved the conjecture proposed by Gorgol in [I. Gorgol. Tur\'{a}n numbers for disjoint copies of graphs. {\it Graphs Combin.}, 27 (2011) 661-667.].

L(2,1)-labeling of interval graphs

Abstract: An L(2,1)-labeling of a graph G = (V, E) is a function f from the vertex set V (G) to the set of non-negative integers such that adjacent vertices get numbers at least two apart, and vertices at distance two get distinct numbers. The L(2,1)- labeling number denoted by λ2,1(G) of G is the minimum range of labels over all such labeling. In this article, it is shown that, for an interval graph G, the upper bound of λ2,1(G)is � +ω,whereand ωrepresentsthemaximumdegreeoftheverticesand size of maximum clique respectively. An O(m + n) time algorithm is also designed to L(2,1)-label a connected interval graph, where m and n represent the number of edges and vertices respectively. Extending this idea it is shown that λ2,1(G) ≤ � +3ω for circular-arc graph.

On Laplacian{Energy{Like Invariant

Abstract: For a simple connected graph G of order n, the Laplacian{energy{like invariant and the Kirchho index are calculated by LEL(G) = n 1 P i=1 p i and Kf(G) = n n 1 P i=1 1= i, respectively, where 1; 2;:::; n 1; n = 0 are the Laplacian eigenvalues of G. We obtain a sharp upper bound forKf and a sharp lower bound forLEL. Further, we obtain upper and lower bounds for LEL and Kf for graphs C(G) (the clique{inserted graph or para-line graph), R(G) (obtained by changing each edge of G into a triangle), and H(G) (obtained by inserting a new vertex on each edge of G and by joining two new vertices if they lie on adjacent edges of G), as well as for the line graph of a semiregular graph.

On characterizing terrain visibility graphs

Abstract: A terrain is an $x$-monotone polygonal line in the $xy$-plane. Two vertices of a terrain are mutually visible if and only if there is no terrain vertex on or above the open line segment connecting them. A graph whose vertices represent terrain vertices and whose edges represent mutually visible pairs of terrain vertices is called a terrain visibility graph . We would like to find properties that are both necessary and sufficient for a graph to be a terrain visibility graph; that is, we would like to characterize terrain visibility graphs. Abello et al. [Discrete and Computational Geometry, 14(3):331--358, 1995] showed that all terrain visibility graphs are “persistent”. They showed that the visibility information of a terrain point set implies some ordering requirements on the slopes of the lines connecting pairs of points in any realization, and as a step towards showing sufficiency, they proved that for any persistent graph $M$ there is a total order on the slopes of the (pseudo) lines in a generalized configuration of points whose visibility graph is $M$. We give a much simpler proof of this result by establishing an orientation to every triple of vertices, reflecting some slope ordering requirements that are consistent with $M$ being the visibility graph, and prove that these requirements form a partial order. We give a faster algorithm to construct a total order on the slopes. Our approach attempts to clarify the implications of the graph theoretic properties on the ordering of the slopes, and may be interpreted as defining properties on an underlying oriented matroid that we show is a restricted type of $3$-signotope.

Spectral study of the Geometric-Arithmetic Index

Abstract: The concept of geometric{arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. One of the main aims of algebraic graph theory is to determine how, or whether, properties of graphs are reected in the algebraic properties of some matrices. The aim of this paper is to study the geometric{arithmetic index GA1 from an algebraic viewpoint. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix that is a modication of the classical adjacency matrix involving the degrees of the vertices.

Decompositions of edge-coloured infinite complete graphs into monochromatic paths II

Abstract: An $r$-edge coloring of a graph or hypergraph $G=(V,E)$ is a map $c:E\to \{0, \dots, r-1\}$. Extending results of Rado and answering questions of Rado, Gy\'arf\'as and S\'ark\"ozy we prove that (1.) the vertex set of every $r$-edge colored countably infinite complete $k$-uniform hypergraph can be partitioned into $r$ monochromatic tight paths with distinct colors (a tight path in a $k$-uniform hypergraph is a sequence of distinct vertices such that every set of $k$ consecutive vertices forms an edge), (2.) for all natural numbers $r$ and $k$ there is a natural number $M$ such that the vertex set of every $r$-edge colored countably infinite complete graph can be partitioned into $M$ monochromatic $k^{th}$ powers of paths apart from a finite set (a $k^{th}$ power of a path is a sequence $v_0, v_1, \dots$ of distinct vertices such that $1\le|i-j| \le k$ implies that $v_iv_j$ is an edge), (3.) the vertex set of every $2$-edge colored countably infinite complete graph can be partitioned into $4$ monochromatic squares of paths, but not necessarily into $3$, (4.) the vertex set of every $2$-edge colored complete graph on $\omega_1$ can be partitioned into $2$ monochromatic paths with distinct colors.

Mapping Simple Polygons: The Power of Telling Convex from Reflex

Abstract: We consider the exploration of a simple polygon P by a robot that moves from vertex to vertex along edges of the visibility graph of P. The visibility graph has a vertex for every vertex of P and an edge between two vertices if they see each other—that is, if the line segment connecting them lies inside P entirely. While located at a vertex, the robot is capable of ordering the vertices it sees in counterclockwise order as they appear on the boundary, and for every two such vertices, it can distinguish whether the angle between them is convex (l π) or reflex ( > π). Other than that, distant vertices are indistinguishable to the robot. We assume that an upper bound on the number of vertices is known.We obtain the general result that a robot exploring any locally oriented, arc-labeled graph G can always determine the base graph of G. Roughly speaking, this is the smallest graph that cannot be distinguished by a robot from G by its observations alone, no matter how it moves. Combining this result with various other techniques allows the ability to show that a robot exploring a polygon P with the preceding capabilities is always capable of reconstructing the visibility graph of P. We also show that multiple identical, indistinguishable, and deterministic robots of this kind can always solve the weak rendezvous problem in which they need to position themselves such that they mutually see each other—for instance, such that they form a clique in the visibility graph.

The Minimum Wiener Connector.

Abstract: The Wiener index of a graph is the sum of all pairwise shortest-path distances between its vertices. In this paper we study the novel problem of finding a minimum Wiener connector: given a connected graph $G=(V,E)$ and a set $Q\subseteq V$ of query vertices, find a subgraph of $G$ that connects all query vertices and has minimum Wiener index. We show that The Minimum Wiener Connector admits a polynomial-time (albeit impractical) exact algorithm for the special case where the number of query vertices is bounded. We show that in general the problem is NP-hard, and has no PTAS unless $\mathbf{P} = \mathbf{NP}$. Our main contribution is a constant-factor approximation algorithm running in time $\widetilde{O}(|Q||E|)$. A thorough experimentation on a large variety of real-world graphs confirms that our method returns smaller and denser solutions than other methods, and does so by adding to the query set $Q$ a small number of important vertices (i.e., vertices with high centrality).

The Book Thickness of 1-Planar Graphs is Constant

Abstract: In a book embedding, the vertices of a graph are placed on the spine of a book and the edges are assigned to pages, so that edges on the same page do not cross. In this paper, we prove that every $1$-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant number of pages. To the best of our knowledge, the best non-trivial previous upper-bound is $O(\sqrt{n})$, where $n$ is the number of vertices of the graph.

Improved Approximation Algorithms for Unsplittable Flow on a Path with Time Windows

Abstract: In the well-studied Unsplittable Flow on a Path problem (UFP), we are given a path graph with edge capacities. Furthermore, we are given a collection of n tasks, each one characterized by a subpath, a weight, and a demand. Our goal is to select a maximum weight subset of tasks so that the total demand of selected tasks using each edge is upper bounded by the corresponding capacity. Chakaravarthy et al. [ESA’14] studied a generalization of UFP, bagUFP, where tasks are partitioned into bags, and we can select at most one task per bag. Intuitively, bags model jobs that can be executed at different times (with different duration, weight, and demand). They gave a \(O(\log n)\) approximation for bagUFP. This is also the best known ratio in the case of uniform weights. In this paper we achieve the following main results:

Fully Dynamic Recognition of Proper Circular-Arc Graphs

Abstract: We present a fully dynamic algorithm for the recognition of proper circular-arc (PCA) graphs. The allowed operations on the graph involve the insertion and removal of vertices (together with its incident edges) or edges. Edge operations cost O(logn) time, where n is the number of vertices of the graph, while vertex operations cost O(logn+d) time, where d is the degree of the modified vertex. We also show incremental and decremental algorithms that work in O(1) time per inserted or removed edge. As part of our algorithm, fully dynamic connectivity and co-connectivity algorithms that work in O(logn) time per operation are obtained. Also, an O(Δ) time algorithm for determining if a PCA representation corresponds to a co-bipartite graph is provided, where Δ is the maximum among the degrees of the vertices. When the graph is co-bipartite, a co-bipartition of each of its co-components is obtained within the same amount of time. As an application, we show how to find a minimal forbidden induced subgraph of a static graph in O(n+m) time.

Edge-partitioning a graph into paths: beyond the Bar\'at-Thomassen conjecture

Abstract: The Barat-Thomassen conjecture asserts that there is a function $f$ such that for every fixed tree $T$ with $t$ edges, every graph which is $f(t)$-edge-connected with its number of edges divisible by $t$ has a partition of its edges into copies of $T$. This has been proved in the case of paths of length $2^k$ by Thomassen, and recently shown to be true for all paths by Botler, Mota, Oshiro and Wakabayashi. Our goal in this paper is to propose an alternative proof of the path case with a weaker hypothesis: Namely, we prove that there is a function $f$ such that every $24$-edge-connected graph with minimum degree $f(t)$ has an edge-partition into paths of length $t$ whenever $t$ divides the number of edges. We also show that $24$ can be dropped to $4$ when the graph is eulerian.

On the construction of increasing-chord graphs on convex point sets

Abstract: A geometric path from s to t is increasing-chord, if while traversing it from s to t the distance to the following (resp. from the preceding) points of the path decreases (resp. increases). A geometric graph is increasing-chord if each two distinct vertices are connected with an increasing-chord path. We show that given a convex point set P in the plane we can construct an increasing-chord graph consisting of P, at most one Steiner point and at most 4|P| — 8 edges.