Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInterval k-Graphs and Orders
88
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
An interval $k$-graph is the intersection graph of a family $mathcal{I}$ of intervals of the real line partitioned into at most $k$ classes with vertices adjacent if and only if their corresponding intervals intersect and belong to different classes. In this paper we discuss the interval $k$-graphs that are the incomparability graphs of orders; i.e., cocomparability interval $k$-graphs or interval $k$-orders. Interval $2$-orders have been characterized in many ways, but we show that analogous characterizations do not carry over to interval $k$-orders, for $k > 2$. We describe the structure of interval $k$-orders, for any $k$, characterize the interval $3$-orders (cocomparability interval $3$-graphs) via one forbidden suborder (subgraph), and state a conjecture for interval $k$-orders (any $k$) that would characterize them via two forbidden suborders.
rate research
Read More
We prove that the order of an ordered group is an interval order if and only if it is a semiorder. Next, we prove that every semiorder is isomorphic to a collection $mathcal J$ of intervals of some totally ordered abelian group, these intervals being of the form $[x, x+ alpha[$ for some positive $alpha$. We describe ordered groups such that the ordering is a semiorder and we introduce threshold groups generalizing totally ordered groups. We show that the free group on finitely many generators and the Thompson group $mathbb F$ can be equipped with a compatible semiorder which is not a weak order. On another hand, a group introduced by Clifford cannot.
We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph if it has a multiple interval representation with at most n+k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k=0), we parameterize graph problems by k, and find FPT algorithms for several problems, including Feedback Vertex Set, Dominating Set, Independent Set, Clique, Clique Cover, and Multiple Interval Transversal. The Coloring problem turns out to be W[1]-hard and we design an XP algorithm for the recognition problem.
Rabinovitch showed in 1978 that the interval orders having a representation consisting of only closed unit intervals have order dimension at most 3. This article shows that the same dimension bound applies to two other classes of posets: those having a representation consisting of unit intervals (but with a mixture of open and closed intervals allowed) and those having a representation consisting of closed intervals with lengths in ${0,1}$.
In this paper we extend the work of Rautenbach and Szwarcfiter by giving a structural characterization of graphs that can be represented by the intersection of unit intervals that may or may not contain their endpoints. A characterization was proved independently by Joos, however our approach provides an algorithm that produces such a representation, as well as a forbidden graph characterization.
We confirm the equitable $Delta$-coloring conjecture for interval graphs and establish the monotonicity of equitable colorability for them. We further obtain results on equitable colorability about square (or Cartesian) and cross (or direct) products of graphs.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
