Let $G$ be a simple connected graph. In this paper, Szeged dimension and PI$_v$ dimension of graph $G$ are introduced. It is proved that if $G$ is a graph of Szeged dimension $1$ then line graph of $G$ is 2-connected. The dimensions of five composite graphs: sum, corona, composition, disjunction and symmetric difference with strongly regular components is computed. Also explicit formulas of Sze...

An interval t coloring of a graph is a proper edgecoloring of with colors 1, such that at least one edge of G is colored by and the edges incident to each vertex − G G 2, , t ... , 1, 2, , , i i t = ... ( ) v V G ∈ are colored by consecutive colors, where is the degree of the vertex in . In this paper interval edge-colorings of various graph products are investigated. ( ) G d v ( ) G d v v G

We consider the problem of partitioning interval graphs into cliques of bounded size. Each interval has a weight, and the cost of a clique is the maximum weight of any interval in the clique. This natural graph problem can be interpreted as a batch scheduling problem. Solving an open question from [7, 4, 5], we show NP-hardness, even if the bound on the clique sizes is constant. Moreover, we gi...

We survey a number of minor-monotone graph parameters and their relationship to the complexity of routing on graphs. In particular we compare the interval routing parameters κslir(G) and κsir(G) with Colin de Verdière’s graph invariant μ(G) and its variants λ(G) and κ(G). We show that for all the known characterizations of θ(G) with θ(G) being μ(G), λ(G) or κ(G), that θ(G) ≤ 2κslir(G) − 1 and θ...

The niche graph of a digraph D is the (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if N D (x)∩N + D (y) 6= ∅ or N − D (x)∩ N D (y) 6= ∅, where N + D (x) (resp. N − D (x)) is the set of out-neighbors (resp. in-neighbors) of x in D. A digraph D = (V,A) is called a semiorder (or a unit interval order) if there exist ...