Intersections of Graphs and \({\chi }\)-Boundedness

iv 1 Preliminaries 1 2 Odd Open Dominating Sets in the Direct Product of Graphs 4 2.1 Odd Open Dominating Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Direct Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Odd Closed r-Dominating Sets in Strong Products of Graphs 11 3.1 Odd Closed r-Dominating Sets . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The Strong Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 The Problem of Enumeration 20 Bibliography 24 Vita 26 Abstract PARITY DOMINATION IN PRODUCT GRAPHS By Christopher Alan Whisenant, Master of Science. A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. Virginia Commonwealth University, 2011. Director: Dewey T. Taylor, Associate Professor, Department of Mathematics and Applied Mathematics.PARITY DOMINATION IN PRODUCT GRAPHS By Christopher Alan Whisenant, Master of Science. A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. Virginia Commonwealth University, 2011. Director: Dewey T. Taylor, Associate Professor, Department of Mathematics and Applied Mathematics. An odd open dominating set of a graph is a subset of the graph’s vertices with the property that the open neighborhood of each vertex in the graph contains an odd number of vertices in the subset. An odd closed r-dominating set is a subset of the graph’s vertices with the property that the closed r-ball centered at each vertex in the graph contains an odd number of vertices in the subset. We first prove that the n-fold direct product of simple graphs has an odd open dominating set if and only if each factor has an odd open dominating set. Secondly, we prove that the n-fold strong product of simple graphs has an odd closed r-dominating set if and only if each factor has an odd closed r-dominating set.

A vague graph is a summed up construction of a fuzzy graph that gives more accuracy to framework when contrasted and frameworks that are planned utilizing fuzzy graphs. In this paper, we consider star related vague graphs G1 and G2 and their cartesian product graph. We find the self-centered vertex and antipodal vague graphs of the product graph, and we investigate the isomorphic property of cartesian product of star related vague graphs. Also we newly introduced the concept of self-corner vertex for the graph and cartesian product of graph.

The problem of finding a fixed point of a nonexpansive mapping on a hypercube is that it has a polynomial time algorithm. In fact, it is known that one can find a 2-satisfiability characterization of the set of all fixed points in polynomial time. This implies that the problem of finding a vertex that is a common fixed point of several given nonexpansive mappings on a hypercube is that it has a polynomial time algorithm. We consider the problem of finding a vertex that is a common fixed point of several given nonexpansive mappings on a more general Cartesian product of graphs. For a single nonexpansive mapping, a known polynomial time algorithm finds a fixed point and a 2-satisfiability-like characterization of all fixed points. We introduce graphs with a farthest point property (also called apiculate graphs in [H. J. Bandelt and V. Chepoi, The Algebra of Metric Betweenness: Subdirect Representations, Retracts, and Axiomatics, manuscript]), and show that finding a common fixed point of several nonexpansive mappings on Cartesian products of such graphs involves using a polynomial time algorithm. We generalize this result to any family of graphs having a majority function. By contrast, the smallest graph (in the sense of having the fewest vertices, and the fewest edges of those having the fewest vertices) without the farthest point property is K2,3 , and finding a vertex that is a fixed point of two given nonexpansive mappings (retractions) on a Cartesian product of graphs isomorphic to K2,3 is NP-complete. More generally, we exhibit an infinite family of graphs without the farthest point property giving NP-completeness. We show that for any family of graphs not having a majority function, the existence of a common fixed point of two nonexpansive mappings on Cartesian products of such graphs is NP-complete. This proves a dichotomy for the problem based on the existence of a majority function; a similar dichotomy is obtained for the special case of nonexpansive mappings that are retractions. Finally we characterize the families of chordal graphs corresponding to both dichotomies.

Disk intersection (respectively, touching) graphs are the intersection graphs of closed disks in the plane whose interiors may (respectively, may not) overlap. In a previous paper [BK93], we showed that the recognition problem for unit disk intersection graphs (i.e. intersection graphs of unit disks) is NP-hard. That proof is easily modified to apply to unit disk touching graphs as well. In this paper, we show how to generalize our earlier construction to accomodate disks whose size may differ. In particular, we prove that the recognition problems for both bounded-ratio disk intersection graphs and bounded-ratio disk touching graphs are also NP-hard. (By bounded-ratio we refer to the natural generalization of the unit constraint in which the radius ratio of the largest to smallest permissible disk is bounded by some fixed constant.) The latter result contrasts with the fact that the disk touching graphs (of unconstrained ratio) are precisely the planar graphs, and are hence polynomial time recognizable. The recognition problem for disk intersection graphs (of unconstrained ratio) has recently been shown to be NP-hard as well [Kra95].

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G1×G2 is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs).

Based on complete bipolar anti fuzzy graphs and on strong bipolar anti fuzzy graphs we can define a new graph through union, join, Cartesian product, and composition of such two graphs. We call the new graph as a product of graphs. In this article we investigate properties of product of bipolar anti fuzzy graphs and product of strong bipolar anti fuzzy graphs. We also construct the degree of a vertex in such the product, namely, the degree of a vertex in the graph which are obtained from product of two given bipolar anti fuzzy graph.

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes.Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.

Fiber-complemented graphs — I: structure and invariant subgraphs

Cartesian Products of Graphs and Their Crossing Numbers

In 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

A graph G is k-factor-critical if G ? S has a perfect matching for any k-subset S of V(G). In this paper, we investigate the factor-criticality in Cartesian products of graphs and show that Cartesian product of an m-factor-critical graph and an n-factor-critical graph is $${(m+n+\varepsilon )}$$ -factor-critical, where $${\varepsilon = 0}$$ if both of m and n are even; $${\varepsilon = 1}$$ , otherwise. Moreover, this result is best possible.

Zip product was recently used in a note establishing the crossing number of the Cartesian product K1,n □ Pm. In this article, we further investigate the relations of this graph operation with the crossing numbers of graphs. First, we use a refining of the embedding method bound for crossing numbers to weaken the connectivity condition under which the crossing number is additive for the zip product. Next, we deduce a general theorem for bounding the crossing numbers of (capped) Cartesian product of graphs with trees, which yields exact results under certain symmetry conditions. We apply this theorem to obtain exact and approximate results on crossing numbers of Cartesian product of various graphs with trees. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 287–300, 2007

Linear formation control is the most simple formation control strategy based on relative states. However, as the number of agents becomes larger, the formation graph becomes more complex and the computation will be increased. The Cartesian product of graphs and the factorization lemma indicates that a complex, connected graph can be decomposed as a Cartesian product of smaller graphs and the system dynamics will thus be simplified by this factorization approach. This work is an extension of the linear formation control strategy and has been examined effective.

For k ∈ Z and G a simple connected graph, a k-radio labeling f : VG → Z of G requires all pairs of distinct vertices u and v to satisfy |f(u)−f(v)| ≥ k+1−d(u, v). When k = 1, this requirement gives rise to the familiar labeling known as vertex coloring for which each vertex of a graph is labeled so that adjacent vertices have different “colors”. We consider k-radio labelings of G when k = diam(G). In this setting, no two vertices can have the same label, so graphs that have radio labelings of consecutive integers are one extreme on the spectrum of possibilities; graphs that can be labeled with such a labeling are called radio graceful. In this thesis, we give four main results on the existence of radio graceful graphs, which focus on Hamming graphs (Cartesian products of complete graphs) and a generalization of the Petersen graph. In particular, we prove the existence of radio graceful graphs of arbitrary diameter, a result previously unknown. Two of these main results show that, under certain conditions, the t Cartesian power G of a radio graceful graph G is also radio graceful. We will also speak to occasions when G is not radio graceful despite G being so, as well as some partial results about necessary and sufficient conditions for a graph G so that G is radio graceful.

Computer or communication networks are so designed that they do not easily get disrupted under external attack and, moreover, these are easily reconstructible if they do get disrupted. These desirable properties of networks can be measured by various parameters like connectivity, toughness, integrity, and tenacity. In an article by Cozzens et al., the authors defined the tenacity of a graph G(V,E) as min {|S| + τ(G −S)/ω(G −S) :S ⊆V}, where τ(G −S) and ω(G −S), respectively, denote the order of the largest component and number of components in G −S. This is a better parameter to measure the stability of a network G, as it takes into account both the quantity and order of components of the graph G −S. The Cartesian products of graphs like hypercubes, grids, and tori are widely used to design interconnection networks in multiprocessor computing systems. These considerations motivated us to study tenacity of Cartesian products of graphs. In this paper, we find the tenacity of Cartesian product of complete graphs (thus settling a conjecture stated in Cozzens et al.) and grids. © 1999 John Wiley & Sons, Inc. Networks 34: 192–196, 1999