Labeled graphs, in which edges are labeled by letters from some alphabet Σ, are extensively used to model many types of relations associated with actions, costs, owners, or other properties. Each path in a labeled graph induces a word in Σ∗ – the one obtained by concatenating the letters along the edges in the path. Classical graph-theory problems give rise to new problems that take these words...

Symbolic model checking is PSPACE complete. Since QBF is the standard PSPACE complete problem, it is most natural to encode symbolic model checking problems as QBF formulas and then use QBF decision procedures to solve them. We discuss alternative encodings for unbounded and bounded safety checking into SAT and QBF. One contribution is a linear encoding of simple path constraints, which usually...

A path in an edge colored graph is said to be a rainbow path if every edge in this path is colored with the same color. The rainbow connection number of G, denoted by rc(G), is the smallest number of colors needed to color its edges, so that every pair of its vertices is connected by at least one rainbow path. A rainbow u − v geodesic in G is a rainbow path of length d(u, v), where d(u, v) is t...

A graph is an X-graph of Y-graphs (or two-level clustered graph) if its vertices can be partitioned into subsets (called clusters) such that each cluster induces a graph belonging to the given class Y and the graph of the clusters belongs to another given class X. Two-level clustered graphs are a useful and interesting concept in graph drawing. We consider the complexity of recognizing two-leve...

the chromatic number of a graph g, denoted by χ(g), is the minimum number of colors such that g can be colored with these colors in such a way that no two adjacent vertices have the same color. a clique in a graph is a set of mutually adjacent vertices. the maximum size of a clique in a graph g is called the clique number of g. the turán graph tn(k) is a complete k-partite graph whose partition...

A set $Wsubseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,vin V(G)$ there exists $win W$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $dim(G)$. In this paper, it is proved that in a connected graph $...

A Hamiltonian cycle (path) of a graph G is a simple cycle (path) which contains all the vertices of G. The Hamiltonian cycle problem asks whether a given graph contains a Hamiltonian cycle. It is NP-complete even for 3-connected planar graphs [3, 61. However, the problem becomes polynomial-time solvable for Cconnected planar graphs: Tutte proved that such a graph necessarily contains a Hamilton...

The traveling–salesman problem (TSP) is one of the most studied problems in combinatorial optimization [16, 14]. The TSP is simply stated, has practical applications, and is a representative of a large class of important scientific and engineering problems. The TSP can be viewed as a graph–theory problem if the cities are identified with the vertices of a graph, and the links between the cities...

A binary matrix satisfies the consecutive ones property (C1P) if its columns can be permuted such that the 1s in each row of the resulting matrix are consecutive. Equivalently, a family of sets F = {Q1, . . . , Qm}, where Qi ⊆ R for some universe R, satisfies the C1P if the symbols in R can be permuted such that the elements of each set Qi ∈ F occur consecutively, as a contiguous segment of the...

background: in recent years, there has been an increase in the amount and variety of data generated in the field of healthcare, (e.g., data related to the prevalence of contagious diseases in the society). various patterns of individuals’ relationships in the society make the analysis of the network a complex, highly important process in detecting and preventing the incidence of diseases. there...