For two xed graphs X and Y the X Y intersection graph of a graph G is a graph whose vertices are induced subgraphs of G isomor phic to Y and where two vertices are adjacent if their intersection in G contains an induced subgraph isomorphic to X A conformal k graph is a simple hypergraph whose hyperedges are exactly k cliques in the section of the hypergraph Let kX denote the disjoint union of k...

Let Γ=(V,E) be a graph and ‎W_(‎a)={w_1,…,w_k } be a subset of the vertices of Γ and v be a vertex of it. The k-vector r_2 (v∣ W_a)=(a_Γ (v,w_1),‎…‎ ,a_Γ (v,w_k)) is the adjacency representation of v with respect to W in which a_Γ (v,w_i )=min{2,d_Γ (v,w_i )} and d_Γ (v,w_i ) is the distance between v and w_i in Γ. W_a is called as an adjacency resolving set for Γ if distinct vertices of ...

For a given graph G and a positive integer k the Pk-path graph Pk(G) has for vertices the set of paths of length k in G. Two vertices are connected in Pk(G) when the intersection of the corresponding paths forms a path of length k − 1 in G, and their union forms either a cycle or a path of length k + 1. Path graphs were proposed as a generalization of line graphs. In this article we investigate...

A graph is Bk-VPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B3-VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B2-VPG. We also show that the 4-connected planar graphs constitute a subclass of the intersection graphs of Z-shapes (i.e., a...

In recent years, determining the common information privately and efficiently between two mutually mistrusting parties have become an important issue in social networks. Many Private set intersection (PSI) protocols have been introduced to address this issue. By applying these protocols, two parties can compute the intersection between their sets without disclosing any information about compone...

Generalizing a decision problem for bipartite perfect matching, Edmonds (J. Res. Natl. Bur. Standards 718(4) (1967) 242) introduced the problem (now known as the Edmonds Problem) of deciding if a given linear subspace ofM(N) contains a non-singular matrix, whereM(N) stands for the linear space of complexN ×N matrices. This problem led to many fundamental developments in matroid theory, etc. Cla...

Two characterizations of hypercubes are given: 1) A graph is a hypercube if and only if it is antipodal and bipartite (0, 2)-graph. 2) A graph is an nhypercube if and only if there are n pairs of prime convexes, the graph is a prime convex intersection graph, and each intersection of n prime convexes (no one of which is from the same pair) is a vertex.

The thinness of a graph is width parameter that generalizes some properties interval graphs, which are exactly the graphs one. Graphs with at most two include, for example, bipartite convex graphs. Many NP-complete problems can be solved in polynomial time bounded thinness, given suitable representation graph. Proper defined analogously, generalizing proper and larger family known to polynomial...