We collect some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes defined by forbidden subgraphs and some classes of intersection graphs, including triangle-free graphs, chordal graphs, cographs, interval graphs, unit interval graphs, threshold graphs, and line graphs.

A uniquely restricted matching is defined to be a matching M whose matched vertices induces a sub-graph which has only one perfect matching. In this paper, we make progress on the open question of the status of this problem on interval graphs (graphs obtained as the intersection graph of intervals on a line). We give an algorithm to compute maximum cardinality uniquely restricted matchings on c...

Interval and proper interval graphs are very well-known graph classes, for which there is a wide literature. As consequence, some generalizations of have been proposed, in general expressed terms $k$ graphs, by splitting the special way. recent example such an approach, classes $k$-thin introduced generalizing respectively. The complexity recognition each these still open, even fixed $k \geq 2$...

Interval graph is a very important subclass of intersection graphs and perfect graphs. It has many applications in different real life situations. The problems on interval graph are solved by using different data structures among them interval tree is very useful. During last decade this data structure is used to solve many problems on interval graphs due to its nice properties. Some of its imp...

We characterize graphs that have intersection representations using unit intervals with open or closed ends such that all ends of the intervals are integral in terms of infinitely many minimal forbidden induced subgraphs. Furthermore, we provide a quadratic-time algorithm that decides if a given interval graph admits such an intersection representation.

In this paper we obtain several characterizations of the adjacency matrix of a probe interval graph. In course of this study we describe an easy method of obtaining interval representation of an interval bipartite graph from its adjacency matrix. Finally, we note that if we add a loop at every probe vertex of a probe interval graph, then the Ferrers dimension of the corresponding symmetric bipa...

Given a graph G with nonnegative node labels w, a multiset of stable sets S1, . . . , Sk ⊆ V (G) such that each vertex v ∈ V (G) is contained in w(v) many of these stable sets is called a weighted coloring. The weighted coloring number χw(G) is the smallest k such that there exist stable sets as above. We provide a polynomial time combinatorial algorithm that computes the weighted coloring numb...

The Induced Subtree Isomorphism problem takes as input a graph G and a tree T , and the task is to decide whether G has an induced subgraph that is isomorphic to T . This problem is known to be NP-complete on bipartite graphs, but it can be solved in polynomial time when G is a forest. We show that Induced Subtree Isomorphism can be solved in polynomial time when G is an interval graph. In cont...