Discrete realizations of contact and intersection graphs (extended abstract)

Fiber-complemented graphs — I: structure and invariant subgraphs

Networks derived from the hypercube are noted for their efficiency in simulating any bounded-degree communication network and hence they play a significant role in parallel machines. For a graph [Formula: see text], a geodetic cover is a subset [Formula: see text] of [Formula: see text], satisfying the condition that every vertex of [Formula: see text] other than the vertices in [Formula: see text] are covered using geodesics connecting any two vertices in [Formula: see text]. The geodetic number of [Formula: see text] is the least order of such covers in [Formula: see text]. This seminal distance-based parameter is well employed in the areas of location theory, convexity theory and game theory. The strong variant and its edge version are recent variations of the problem that were introduced in due of their applications in social networks. The most established and strong hypercube derivative networks are studied in this paper with respect to certain geodetic parameters. Notably, among the networks that we have considered, the butterfly and the Benes networks belong to the class of bipartite graphs for which the geodetic number problem and its strong version are NP-complete. We have determined the geodetic parameters that include the geodetic number, the strong geodetic number and their corresponding edge variants of the networks taken into consideration. In the process, we have identified a new class of graphs, as semi-extreme edge geodesic graphs and that the enhanced butterfly networks belong to this class. The study of such geodetic parameters in the structural models of parallel computation could be useful for the design of efficient algorithms for the structures considered.

Let A be a convex body in the plane and A₁,…,A_n be translates of A. Such translates give rise to an intersection graph of A, G = (V,E), with vertices V = {1,… ,n} and edges E = {uv∣ A_u ∩ A_v ≠ ∅}. The subgraph G' = (V, E') satisfying that E' ⊂ E is the set of edges uv for which the interiors of A_u and A_v are disjoint is a unit distance graph of A. If furthermore G' = G, i.e., if the interiors of A_u and A_v are disjoint whenever u≠ v, then G is a contact graph of A. In this paper, we study which pairs of convex bodies have the same contact, unit distance, or intersection graphs. We say that two convex bodies A and B are equivalent if there exists a linear transformation B' of B such that for any slope, the longest line segments with that slope contained in A and B', respectively, are equally long. For a broad class of convex bodies, including all strictly convex bodies and linear transformations of regular polygons, we show that the contact graphs of A and B are the same if and only if A and B are equivalent. We prove the same statement for unit distance and intersection graphs.

In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time. Since the recognition problem for this class of graphs is NP-hard we can not rely on standard techniques for geometric graphs based on constructing a canonical realization. Instead, we develop new techniques which combine structural insights into the class of unit square graphs with understanding of the automorphism group of such graphs. For the latter we introduce a generalization of bounded degree graphs which is used to capture the main structure of unit square graphs. Using group theoretic algorithms we obtain sufficient information to solve the isomorphism problem for unit square graphs.

Classes and Recognition of Curve Contact Graphs

The metric representation of a vertex v of a graph G is a finite vector representing distances of v with respect to vertices of some ordered subset S⊆V (G). If no suitable subset of S provides separate representations for each vertex of V(G), then the set S is referred to as a minimal resolving set. The metric dimension of G is the cardinality of the smallest (with respect to its cardinality) minimal resolving set. A resolving set S is secure if for any v∈V–S, there exists x∈S such that (S–{x})∪{v} is a resolving set. For various classes of graphs, the value of the secure resolving number is determined and defined. The secure metric dimension of the graph classes is being studied in this work. The results show that different graph families have different metric dimensions.

Cops and Robbers on intersection graphs

Cover-Incomparability graphs (C-I graphs) form an interesting class of graphs from posets. C-I graphs are identified among chordal graphs, distance-hereditary graphs, Ptolemaic graphs, split graphs, threshold graphs, bisplit graphs, block graphs and cographs. Thus only a few classes of graphs are known to be C-I graphs so far. Composition operation and various graph products are usually used to produce more non-trivial graphs in a particular graph class, using the prime graphs in the class. In this paper, we attempt to study the effect of the composition, lexicographic and strong products of C-I graphs. We found that the lexicographic product of two C-I graphs, say G and H is a C-I graph if and only if either G is any C-I graph, and H is a complete graph or vice versa. A similar result holds for the strong product also. It can be observed that the composition operation is more general than lexicographic product and we obtain new classes of C-I graphs from this operation.

Matthews and Sumner have proved in [10] that if G is a 2-connected claw-free graph of order n such that δ(G) ≥ (n-2) /3, then G is Hamiltonian. We say that a graph is almost claw-free if for every vertex v of G, 〈N(v)〉 is 2-dominated and the set A of centers of claws of G is an independent set. Broersma et al. [5] have proved that if G is a 2-connected almost claw-free graph of order n such that n such that δ(G) ≥ (n-2) /3, then G is Hamiltonian. We generalize these results by considering the graphs satisfying the following property: for every vertex v ∈ A , there exist exactly two vertices x and y of V\A such that N(v) ⊆ N[x] ∪ N[y] . We extend some other known results on claw-free graphs to this new class of graphs.

We construct the first known examples of infinite subgroups of the outer automorphism group of Out(A_Gamma), for certain right-angled Artin groups A_Gamma. This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of Out(Out(A_Gamma)). This demonstrates a marked departure from the known behavior of Out(Out(A_Gamma)) when A_Gamma is free or free abelian, as in these cases Out(Out(A_Gamma)) has order at most 4. We also disprove a previous conjecture of the second author, producing new examples of finite order members of certain Out(Aut(A_Gamma)).

A Semidefinite Programming (SDP) relaxation is an effective computational method to solve a Sensor Network Localization problem, which attempts to determine the locations of a group of sensors given the distances between some of them. In this paper, we analyze and determine new sufficient conditions and formulations that guarantee that the SDP relaxation is exact, i.e., gives the correct solution. These conditions can be useful for designing sensor networks and managing connectivities in practice. Our main contribution is threefold: First, we present the first non-asymptotic bound on the connectivity (or radio) range requirement of randomly distributed sensors in order to ensure the network is uniquely localizable with high probability. Determining this range is a key component in the design of sensor networks, and we provide a result that leads to a correct localization of each sensor, for any number of sensors. Second, we introduce a new class of graphs that can always be correctly localized by an SDP relaxation. Specifically, we show that adding a simple objective function to the SDP relaxation model will ensure that the solution is correct when applied to a triangulation graph. Since triangulation graphs are very sparse, this is informationally efficient, requiring an almost minimal amount of distance information. Finally, we analyze a number of objective functions for the SDP relaxation to solve the localization problem for a general graph.Key wordsSensor network localizationGraph realizationSemidefinite programmingSubject Classifications90C2290C4690C9090B18

Polar graphs and maximal independent sets

In 1983, Klaus studied a class of potentials with bumps and computed the essential spectrum of the associated Schr{\"o}dinger operator with the help of some localisations at infinity. A key hypothesis is that the distance between two consecutive bumps tends to infinity at infinity. In this article, we introduce a new class of graphs (with patterns) that mimics this situation, in the sense that the distance between two patterns tends to infinity at infinity. These patterns tend, in some way, to asymptotic graphs. They are the localisations at infinity. Our result is that the essential spectrum of the Laplacian acting on our graph is given by the union of the spectra of the Laplacian acting on the asymptotic graphs. We also discuss the question of the stability of the essential spectrum in the appendix.

Given a simple connected undirected graph G and let k be the maximum number of its vertices and its edges. Let f be a bijective labeling from the set of its edges to the set of odd integers from 1 up to 2q − 1, where q is the number of edges of G. The labeling f is called an edge odd graceful labeling on G if the weights of any two different vertices are different, where the weight of a vertex v is defined as the sum mod(2k) of all labels of edges that are incident to v. A graph is called an edge odd graceful graph if it admits an edge odd graceful labeling. In this paper, we show that there are some new classes of graphs that are edge odd graceful.

For a simple graph G on n vertices and an integer k with 1 ⩽ k ⩽ n, denote by $$\mathcal{S}^+_k$$ (G) the sum of k largest signless Laplacian eigenvalues of G. It was conjectured that $$\mathcal{S}^+_k(G)\leqslant{e}(G)+(^{k+1}_{\;\;2})$$ (G) ⩽ e(G) + (k+1 2), where e(G) is the number of edges of G. This conjecture has been proved to be true for all graphs when k ∈ {1, 2, n − 1, n}, and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all k). In this note, this conjecture is proved to be true for all graphs when k = n − 2, and for some new classes of graphs.