Random mappings of scaled graphs

Given a connected finite graph Γ with a fixed base point O and some graph G with a based point we study random 1-Lipschitz maps of a scaled Γ into G. We are mostly interested in the case where G is a Cayley graph of some finitely generated group, where the construction does not depend on the choice of base points. A particular case of Γ being a graph on two vertices and one edge corresponds to the random walk on G, and the case where Γ is a graph on two vertices and two edges joining them corresponds to Brownian bridge in G. We show, that unlike in the case ${G=\mathbb Z^d}$ , the asymptotic behavior of a random scaled mapping of Γ into G may differ significantly from the asymptotic behavior of random walks or random loops in G. In particular, we show that this occurs when G is a free non-Abelian group. Also we consider the case when G is a wreath product of ${\mathbb Z}$ with a finite group. To treat this case we prove new estimates for transition probabilities in such wreath products. For any group G generated by a finite set S we define a functor E from category of finite connected graphs to the category of equivalence relations on such graphs. Given a finite connected graph Γ, the value E _G,S (Γ) can be viewed as an asymptotic invariant of G.     

On the well-coveredness of Cartesian products of graphs

A graph G is well-covered if every maximal independent set has the same cardinality. This paper investigates when the Cartesian product of two graphs is well-covered. We prove that if G and H both belong to a large class of graphs that includes all non-well-covered triangle-free graphs and most well-covered triangle-free graphs, then G×H is not well-covered. We also show that if G is not well-covered, then neither is G×G. Finally, we show that G×G is not well-covered for all graphs of girth at least 5 by introducing super well-covered graphs and classifying all such graphs of girth at least 5.     

Assouad–Nagata dimension of connected Lie groups

We prove that the asymptotic Assouad–Nagata dimension of a connected Lie group G equipped with a left-invariant Riemannian metric coincides with its topological dimension of G/C where C is a maximal compact subgroup. To prove it we will compute the Assouad–Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad–Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometrically embedded into any cocompact lattice on a connected Lie group.     

Tree-edges deletion problems with bounded diameter obstruction sets

We study the following problem: given a tree G and a finite set of trees H, find a subset O of the edges of G such that G-O does not contain a subtree isomorphic to a tree from H, and O has minimum cardinality. We give sharp boundaries on the tractability of this problem: the problem is polynomial when all the trees in H have diameter at most 5, while it is NP-hard when all the trees in H have diameter at most 6. We also show that the problem is polynomial when every tree in H has at most one vertex with degree more than 2, while it is NP-hard when the trees in H can have two such vertices.The polynomial-time algorithms use a variation of a known technique for solving graph problems. While the standard technique is based on defining an equivalence relation on graphs, we define a quasiorder. This new variation might be useful for giving more efficient algorithm for other graph problems.     

Partitioning 2-edge-colored graphs by monochromatic paths and cycles

We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from o(|V (G)|) vertices, the vertex set of any 2-edge-colored graph G with minimum degree at least \(\tfrac{{(1 + \varepsilon )3|V(G)|}} {4}\) can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that \(\bar G\) does not contain a fixed bipartite graph H, we show that in every 2-edge-coloring of G, |V (G)| ? c(H) vertices can be covered by two vertex disjoint paths of different colors, where c(H) is a constant depending only on H. In particular, we prove that c(C ₄)=1, which is best possible.     

Spectra of coronae

We introduce a new invariant, the coronal of a graph, and use it to compute the spectrum of the corona G°H of two graphs G and H. In particular, we show that this spectrum is completely determined by the spectra of G and H and the coronal of H. Previous work has computed the spectrum of a corona only in the case that H is regular. We then explicitly compute the coronals for several families of graphs, including regular graphs, complete n-partite graphs, and paths. Finally, we use the corona construction to generate many infinite families of pairs of cospectral graphs.     

On the Metric Dimension of Infinite Graphs

A set of vertices S resolves a connected graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we undertake the metric dimension of infinite locally finite graphs, i.e., those infinite graphs such that all its vertices have finite degree. We give some necessary conditions for an infinite graph to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some general results about the metric dimension of the Cartesian product of finite and infinite graphs, and obtain the metric dimension of the Cartesian product of several families of graphs.     

ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION

In this paper,we are dealing with the study of the metric dimension of some classes of regular graphs by considering a class of bridgeless cubic graphs called the flower snarks Jn,a class of cubic convex polytopes considering the open problem raised in [M.Imran et al.,families of plane graphs with constant metric dimension,Utilitas Math.,in press] and finally Harary graphs H 5,n by partially answering to an open problem proposed in [I.Javaid et al.,Families of regular graphs with constant metric dimension,Utilitas Math.,2012,88:43-57].We prove that these classes of regular graphs have constant metric dimension.     

Shadows of ordered graphs

Isoperimetric inequalities have been studied since antiquity, and in recent decades they have been studied extensively on discrete objects, such as the hypercube. An important special case of this problem involves bounding the size of the shadow of a set system, and the basic question was solved by Kruskal (in 1963) and Katona (in 1968). In this paper we introduce the concept of the shadow ∂G of a collection G of ordered graphs, and prove the following, simple-sounding statement: if n∈N is sufficiently large, |V(G)|=n for each G∈G, and |G|<n, then |∂G|?|G|. As a consequence, we substantially strengthen a result of Balogh, Bollobás and Morris on hereditary properties of ordered graphs: we show that if P is such a property, and |Pk|<k for some sufficiently large k∈N, then |Pn| is decreasing for k?n<∞.     

Induced Graph Packing Problems

Let H{\mathcal{H}} be a set of undirected graphs. The induced H{\mathcal{H}} -packing problem in an input graph G is to find a subgraph Q of G of maximum size such that each connected component of Q is an induced subgraph of G and is isomorphic to some member of H{\mathcal{H}} . In this paper we focus on the case when H{\mathcal{H}} consists of factor-critical graphs and a certain family of ‘propellers’. Clarifying the methods developed in the related theory of non-induced graph packings, we show a Gallai–Edmonds type structure theorem and a Berge–Tutte type minimax formula. We also give an Edmonds type alternating forest algorithm for the case when H{\mathcal{H}} consists of a sequential set of stars and factor-critical graphs. This simplifies the related result of Egawa, Kano and Kelmans.     

Netlike partial cubes I. General properties

A set A of vertices of a graph G is C-convex if the vertex set of any cycle of the subgraph of G induced by the union of the intervals between each pair of elements of A is contained in A. A partial cube (isometric subgraph of a hypercube) is a netlike partial cube if, for each edge ab, the sets U_ab and U_ba are C-convex (U_ab being the set of all vertices closer to a than to b and adjacent to some vertices closer to b than to a, and vice versa for U_ba). Particular netlike partial cubes are median graphs, even cycles, benzenoid graphs and cellular bipartite graphs. In this paper we give different characterizations and properties of netlike partial cubes. In particular, as median graphs and cellular bipartite graphs, these graphs have a pre-hull number which is at most one, and moreover the convex hull of any isometric cycle of a netlike partial cube is, as in the case of bipartite cellular graphs, this cycle itself or, as in the case of median graphs, a hypercube. We also characterize the gated subgraphs of a netlike partial cube, and we show that the gated amalgam of two netlike partial cubes is a netlike partial cube.     

On the Minimum Average Distance Spanning Tree of the Hypercube

Given an undirected and connected graph G, with a non-negative weight on each edge, the Minimum Average Distance (MAD) spanning tree problem is to find a spanning tree of G which minimizes the average distance between pairs of vertices. This network design problem is known to be NP-hard even when the edge-weights are equal. In this paper we make a step towards the proof of a conjecture stated by A.A. Dobrynin, R. Entringer and I. Gutman in 2001, and which says that the binomial tree B _n is a MAD spanning tree of the hypercube H _n . More precisely, we show that the binomial tree B _n is a local optimum with respect to the 1-move heuristic which, starting from a spanning tree T of the hypercube H _n , attempts to improve the average distance between pairs of vertices, by adding an edge e of H _n -T and removing an edge e′ from the unique cycle created by e. We also present a greedy algorithm which produces good solutions for the MAD spanning tree problem on regular graphs such as the hypercube and the torus.     

Two-factors in orientated graphs with forbidden transitions

The instance of the problem of finding 2-factors in an orientated graph with forbidden transitions consists of an orientated graph G and for each vertex v of G, a graph H_v of allowed transitions at v. Vertices of the graph H_v are the edges incident to v. An orientated 2-factor F of G is called legal if all the transitions are allowed, i.e. for every vertex v, the two edges of F adjacent to it form an edge in H_v. Deciding whether a legal 2-factor exists in G is NP-complete in general. We investigate the case when the graphs of allowed transitions are taken from some fixed class C. We provide an exact characterization of classes C so that the problem is NP-complete. In particular, we prove a dichotomy for this problem, i.e. that for any class C it is either polynomial or NP-complete.     

An efficient representation of Benes networks and its applications

The most popular bounded-degree derivative network of the hypercube is the butterfly network. The Benes network consists of back-to-back butterflies. There exist a number of topological representations that are used to describe butterfly—like architectures. We identify a new topological representation of butterfly and Benes networks.The minimum metric dimension problem is to find a minimum set of vertices of a graph G(V,E) such that for every pair of vertices u and v of G, there exists a vertex w with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. It is NP-hard in the general sense. We show that it remains NP-hard for bipartite graphs. The algorithmic complexity status of this NP-hard problem is not known for butterfly and Benes networks, which are subclasses of bipartite graphs. By using the proposed new representations, we solve the minimum metric dimension problem for butterfly and Benes networks. The minimum metric dimension problem is important in areas such as robot navigation in space applications.     

On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x∈V(G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O(mlogn) time whether G is a median graph with geodetic number 2.     

Balanced and 1-balanced graph constructions

There are several density functions for graphs which have found use in various applications. In this paper, we examine two of them, the first being given by b(G)=|E(G)|/|V(G)|, and the other being given by g(G)=|E(G)|/(|V(G)|−ω(G)), where ω(G) denotes the number of components of G. Graphs for which b(H)≤b(G) for all subgraphs H of G are called balanced graphs, and graphs for which g(H)≤g(G) for all subgraphs H of G are called 1-balanced graphs (also sometimes called strongly balanced or uniformly dense in the literature). Although the functions b and g are very similar, they distinguish classes of graphs sufficiently differently that b(G) is useful in studying random graphs, g(G) has been useful in designing networks with reduced vulnerability to attack and in studying the World Wide Web, and a similar function is useful in the study of rigidity. First we give a new characterization of balanced graphs. Then we introduce a graph construction which generalizes the Cartesian product of graphs to produce what we call a generalized Cartesian product. We show that generalized Cartesian product derived from a tree and 1-balanced graphs are 1-balanced, and we use this to prove that the generalized Cartesian products derived from 1-balanced graphs are 1-balanced.     

Parameterized graph cleaning problems

We investigate the Induced Subgraph Isomorphism problem with non-standard parameterization, where the parameter is the difference |V(G)|−|V(H)| with H and G being the smaller and the larger input graph, respectively. Intuitively, we can interpret this problem as “cleaning” the graph G, regarded as a pattern containing extra vertices indicating errors, in order to obtain the graph H representing the original pattern. We show fixed-parameter tractability of the cases where both H and G are planar and H is 3-connected, or H is a tree and G is arbitrary. We also prove that the problem when H and G are both 3-connected planar graphs is NP-complete without parameterization.     

On the complexity of the independent set problem in triangle graphs

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G−I, there is a vertex w∈I such that {u,v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NP-complete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P=NP.     

Backbone colorings of graphs with bounded degree

We study backbone colorings, a variation on classical vertex colorings: Given a graph G and a subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex k-coloring of G in which the colors assigned to adjacent vertices in H differ by at least 2. The minimal k∈N for which such a coloring exists is called the backbone chromatic number of G. We show that for a graph G of maximum degree Δ where the backbone graph is a d-degenerated subgraph of G, the backbone chromatic number is at most Δ+d+1 and moreover, in the case when the backbone graph being a matching we prove that the backbone chromatic number is at most Δ+1. We also present examples where these bounds are attained.Finally, the asymptotic behavior of the backbone chromatic number is studied regarding the degrees of G and H. We prove for any sparse graph G that if the maximum degree of a backbone graph is small compared to the maximum degree of G, then the backbone chromatic number is at most .     

Covariant representations of Hecke algebras and imprimitivity for crossed products by homogeneous spaces

For discrete Hecke pairs (G,H), we introduce a notion of covariant representation which reduces in the case where H is normal to the usual definition of covariance for the action of G/H on c₀(G/H) by right translation; in many cases where G is a semidirect product, it can also be expressed in terms of covariance for a semigroup action. We use this covariance to characterise the representations of c₀(G/H) which are multiples of the multiplication representation on ?²(G/H), and more generally, we prove an imprimitivity theorem for regular representations of certain crossed products by coactions of homogeneous spaces. We thus obtain new criteria for extending unitary representations from H to G.