Transversal numbers of translates of a convex body

Let F be a family of translates of a fixed convex set M in Rⁿ. Let τ(F) and ν(F) denote the transversal number and the independence number of F, respectively. We show that ν(F)?τ(F)?8ν(F)-5 for n=2 and τ(F)?2^n-1nⁿν(F) for n?3. Furthermore, if M is centrally symmetric convex body in the plane, then ν(F)?τ(F)?6ν(F)-3.     

On the total variation distance between the binomial random graph and the random intersection graph

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karoński, Scheinerman, and Singer‐Cohen introduced a random intersection graph by taking randomly assigned sets. The random intersection graph has n vertices and sets assigned to the vertices are chosen to be i.i.d. random subsets of a fixed set M of size m where each element of M belongs to each random subset with probability p, independently of all other elements in M. In 2000, Fill, Scheinerman, and Singer‐Cohen showed that the total variation distance between the random graph and the Erdös‐Rényi graph tends to 0 for any if , where is chosen so that the expected numbers of edges in the two graphs are the same. In this paper, it is proved that the total variation distance still tends to 0 for any whenever .     

Coloring the complements of intersection graphs of geometric figures

Let be the complement of the intersection graph G of a family of translations of a compact convex figure in Rⁿ. When n=2, we show that , where γ(G) is the size of the minimum dominating set of G. The bound on is sharp. For higher dimension we show that , for n?3. We also study the chromatic number of the complement of the intersection graph of homothetic copies of a fixed convex body in Rⁿ.     

The convex dimension of a graph

The convex dimension of a graph G=(V,E) is the smallest dimension d for which G admits an injective map f:V?R^d of its vertices into d-space, such that the barycenters of the images of the edges of G are in convex position. The strong convex dimension of G is the smallest d for which G admits a map as above such that the images of the vertices of G are also in convex position. In this paper we study the convex and strong convex dimensions of graphs.     

Isomorphic properties of intersection bodies

We study isomorphic properties of two generalizations of intersection bodies - the class of k-intersection bodies in Rn and the class of generalized k-intersection bodies in Rn. In particular, we show that all convex bodies can be in a certain sense approximated by intersection bodies, namely, if K is any symmetric convex body in Rn and 1≤k≤n−1 then the outer volume ratio distance from K to the class can be estimated by

     

Some remarks on the geodetic number of a graph

A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D. The geodetic number of G is the minimum cardinality of a geodetic set of G.We prove that it is NP-complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k. Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs.     

Partitioning a graph into convex sets

Let G be a finite simple graph. Let S⊆V(G), its closed interval I[S] is the set of all vertices lying on shortest paths between any pair of vertices of S. The set S is convex if I[S]=S. In this work we define the concept of a convex partition of graphs. If there exists a partition of V(G) into p convex sets we say that G is p-convex. We prove that it is NP-complete to decide whether a graph G is p-convex for a fixed integer p≥2. We show that every connected chordal graph is p-convex, for 1≤p≤n. We also establish conditions on n and k to decide if the k-th power of a cycle C_n is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.     

A lower bound for the interval number of a graph

The interval number i(G) of a graph G with n vertices is the lowest integer m such that G is the intersection graph of some family of sets I₁,…,I_n with every I_i being the union of at most m real intervals. In this article a lower bound for i(G) is proved followed by some considerations about the construction of graphs that are critical with respect to the interval number.     

A remark on intersection of convex sets

Let X be a real Banach space. Let be a family of closed, convex subsets of X. We show that either the intersection ?_γ∈Γ(G_γ)^? of the ?-neighborhood of the sets G_γ is bounded for each ?>0, or it is unbounded for each ?>0. From this we derive a fixed point theorem for suitable maps that move some points along a bounded direction in Hilbert spaces.     

On embedding a cycle in a plane graph

Consider a planar drawing Γ of a planar graph G such that the vertices are drawn as small circles and the edges are drawn as thin stripes. Consider a non-simple cycle c of G. Is it possible to draw c as a non-intersecting closed curve inside Γ, following the circles that correspond in Γ to the vertices of c and the stripes that connect them? We show that this test can be done in polynomial time and study this problem in the framework of clustered planarity for highly non-connected clustered graphs.     

Degree distribution of a typical vertex in a general random intersection graph

For a general random intersection graph, we show an approximation of the vertex degree distribution by a Poisson mixture. Research supported by Lithuanian State Science and Studies Foundation Grant T-07149.     

Computing the maximum clique in the visibility graph of a simple polygon

In this paper, we present an algorithm for computing the maximum clique in the visibility graph G of a simple polygon P in O(n²e) time, where n and e are number of vertices and edges of G respectively. We also present an O(ne) time algorithm for computing the maximum hidden vertex set in the visibility graph G of a convex fan P. We assume in both algorithms that the Hamiltonian cycle in G that corresponds to the boundary of P is given as an input along with G.     

Decomposing a simple polygon into pseudo-triangles and convex polygons

In this paper we consider the problem of decomposing a simple polygon into subpolygons that exclusively use vertices of the given polygon. We allow two types of subpolygons: pseudo-triangles and convex polygons. We call the resulting decomposition PT-convex. We are interested in minimum decompositions, i.e., in decomposing the input polygon into the least number of subpolygons. Allowing subpolygons of one of two types has the potential to reduce the complexity of the resulting decomposition considerably.The problem of decomposing a simple polygon into the least number of convex polygons has been considered. We extend a dynamic-programming algorithm of Keil and Snoeyink for that problem to the case that both convex polygons and pseudo-triangles are allowed. Our algorithm determines such a decomposition in O(n³) time and space, where n is the number of the vertices of the polygon.     

Connectivity of the uniform random intersection graph

A uniform random intersection graphG(n,m,k) is a random graph constructed as follows. Label each of n nodes by a randomly chosen set of k distinct colours taken from some finite set of possible colours of size m. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless sensor networks, in particular when modelling the network graph of the well-known key predistribution technique due to Eschenauer and Gligor.The paper determines the threshold for connectivity of the graph G(n,m,k) when n→∞ in many situations. For example, when k is a function of n such that k≥2 and m=⌊n^α⌋ for some fixed positive real number α then G(n,m,k) is almost surely connected when

lim infk²n/mlogn>1,     

Finding the projection of a point onto the intersection of convex sets via projections onto half-spaces

We present a modification of Dykstra's algorithm which allows us to avoid projections onto general convex sets. Instead, we calculate projections onto either a half-space or onto the intersection of two half-spaces. Convergence of the algorithm is established and special choices of the half-spaces are proposed.The option to project onto half-spaces instead of general convex sets makes the algorithm more practical. The fact that the half-spaces are quite general enables us to apply the algorithm in a variety of cases and to generalize a number of known projection algorithms.The problem of projecting a point onto the intersection of closed convex sets receives considerable attention in many areas of mathematics and physics as well as in other fields of science and engineering such as image reconstruction from projections.In this work we propose a new class of algorithms which allow projection onto certain super half-spaces, i.e., half-spaces which contain the convex sets. Each one of the algorithms that we present gives the user freedom to choose the specific super half-space from a family of such half-spaces. Since projecting a point onto a half-space is an easy task to perform, the new algorithms may be more useful in practical situations in which the construction of the super half-spaces themselves is not too difficult.     

Constant tolerance intersection graphs of subtrees of a tree

A chordal graph is the intersection graph of a family of subtrees of a host tree. In this paper we generalize this. A graph G=(V,E) has an (h,s,t)-representation if there exists a host tree T of maximum degree at most h, and a family of subtrees {S_v}_v∈V of T, all of maximum degree at most s, such that uv∈E if and only if |S_u∩S_v|?t. For given h,s, and t, there exist infinitely many forbidden induced subgraphs for the class of (h,s,t)-graphs. On the other hand, for fixed h?s?3, every graph is an (h,s,t)-graph provided that we take t large enough. Under certain conditions representations of larger graphs can be obtained from those of smaller ones by amalgamation procedures. Other representability and non-representability results are presented as well.