A Graph-Theoretic Barcode Ordering Model for Linked-Reads

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b‐dimensional cube is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b‐dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line—i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ⌈log2ψ(G)⌉. In this article, we show that for an interval graph G ⌈log2ψ(G)⌉⩽cub(G)⩽⌈log2ψ(G)⌉+2. It is not clear whether the upper bound of ⌈log2ψ(G)⌉+2 is tight: till now we are unable to find any interval graph with cub(G)>⌈log2ψ(G)⌉. We also show that for an interval graph G, cub(G)⩽⌈log2α⌉, where α is the independence number of G. Therefore, in the special case of ψ(G)=α, cub(G) is exactly ⌈log2α2⌉. The concept of cubicity can be generalized by considering boxes instead of cubes. A b‐dimensional box is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k‐dimensional boxes. It is clear that box(G)⩽cub(G). From the above result, it follows that for any graph G, cub(G)⩽box(G)⌈log2α⌉. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323–333, 2010

A matrix of zeroes and ones is said to have the consecutive ones property if there is a permutation of its rows such that the ones in each column appear consecutively. This paper develops a data structure which may be used to test a matrix for the consecutive ones property, and produce the desired permutation of the rows, in linear time. One application of the consecutive ones property is in recognizing interval graphs. A graph is an interval graph if there exists a 1-1 correspondence between its vertices and a set of intervals on the real line such that two vertices are adjacent if and only if the corresponding intervals have a nonempty intersection. Fulkerson and Gross have characterized interval graphs as those for which the clique versus vertex incidence matrix has the consecutive ones property. In testing this particular matrix for the consecutive ones property we may process the columns in a special order to simplify the algorithm. This yields the interval graph recognition algorithm which is presented in section 2; section 3 indicates how this algorithm may be extended to the general consecutive ones problem.

Uniquely orderable interval graphs

Two Problems on Interval Counting

Theinterval count problemdetermines the smallest number of interval lengths needed in order to represent an interval model of a given interval graph or interval order. Despite the large number of studies about interval graphs and interval orders, surprisingly only a few results on the interval count problem are known. In this work, we provide a short survey about the interval count and related problems. a graph and the number of its maximal cliques.

The Maximal Independent Set problem is a well-studied problem in the distributed community. We study Maximum and Maximal Independent Set problems on two geometric intersection graphs; interval graphs and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in the local communication model. We compute the maximum independent set on interval graphs in O(k) rounds and O(n) messages, where k is the size of the maximum independent set and n is the number of nodes in the graph. We provide a matching lower bound of \(\varOmega (k)\) on the number of rounds whereas \(\varOmega (n)\) is a trivial lower bound on message complexity. Thus our algorithm is both time and message optimal. We also study the maximal independent set problem in bi-interval graphs, a special case of the interval graphs where the intervals have two different lengths. We prove that a maximal independent set can be computed in bi-interval graphs in constant rounds that is \(\frac{1}{6}\)-approximation. For axis-parallel segment intersection graphs, we design an algorithm that finds a maximal independent set in O(D) rounds, where D is the diameter of the graph. We further show that this independent set is a \(\frac{1}{2}\)-approximation. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].

In this chapter, a very important class of graphs called intersection graph is introduced. Based on the geometrical representation, many different types of intersection graphs can be defined with interesting properties. Some of them—interval graphs, circular-arc graphs, permutation graphs, trapezoid graphs, chordal graphs, line graphs, disk graphs, string graphs—are presented here. A brief introduction of each of these intersection graphs along with some basic properties and algorithmic status are investigated.

Counting endpoint sequences for interval orders and interval graphs

Author(s): Lueker, George S.; Booth, Kellogg S. | Abstract: A graph is an interval graph if and only if each of its vertices can be associated with an interval on the real line in such a way that two vertices are adjacent in the graph exactly when the corresponding intervals have a nonempty intersection. An efficient algorithm for testing isomorphism of interval graphs is implemented using a data structure called a PQ-tree. The algorithm runs in 0(n + e) steps for graphs having n vertices and e edges. It is shown that for a somewhat larger class of graphs, namely the chordal graphs, isomorphism is as hard as for general graphs.

Graph labelling problem has been broadly studied for a long period for its applications, especially in frequency assignment in (mobile) communication system, X -ray crystallography, circuit design, etc. Nowadays, surjective L 2,1 -labelling is a well-studied problem. Motivated from the L 2,1 -labelling problem and the importance of surjective L 2,1 -labelling problem, we consider surjective L 2,1 -labelling ( SL 21 -labelling) problems for paths and interval graphs. For any graph G = V , E , an SL 21 -labelling is a mapping φ : V ⟶ 1,2 , … , n so that, for every pair of nodes u and v , if d u , v = 1 , then φ u − φ v ≥ 2 ; and if d u , v = 2 , then φ u − φ v ≥ 1 , and every label 1,2 , … , n is used exactly once, where d u , v represents the distance between the nodes u and v , and n is the number of nodes of graph G . In the present article, it is proved that any path P n can be surjectively L 2,1 -labelled if n ≥ 4 , and it is also proved that any interval graph IG G having n nodes and degree Δ > 2 can be surjectively L 2,1 -labelled if n = 3 Δ − 1 . Also, we have designed two efficient algorithms for surjective L 2,1 -labelling of paths and interval graphs. The results regarding both paths and interval graphs are the first result for surjective L 2,1 -labelling.

We study the computational complexity of the FO model checking problem on interval graphs, i.e., intersection graphs of intervals on the real line. The main positive result is that this problem can be solved in time O(n logn) for n-vertex interval graphs with representations containing only intervals with lengths from a prescribed finite set. We complement this result by showing that the same is not true if the lengths are restricted to any set that is dense in some open subset, e.g., in the set (1, 1 + ε).

Max point-tolerance graphs

Order, please!: Peter C. Fishburn. Interval orders and interval graphs. Wiley, New York, 1985. pp. xi + 215. $33.50

: There are four prominent product graphs in graph theory: Cartesian, strong, direct and lexicographic. Of these four product graphs, the lexicographic product graph is the least studied. Lexicographic products are not commutative but still have some interesting properties. This paper begins with basic definitions of graph theory, including the definition of a graph, that are needed to understand theorems and proofs that come later. The paper then discusses the lexicographic product of digraphs, denoted G o H, for some digraphs G and H. The paper concludes by proving a cancellation property for the lexicographic product of digraphs G, H, A, and B: if G o H = A o B and /V(G)/ = /V(A)/, then G = A. It also proves additional cancellation properties for lexicographic product digraphs and the author hopes the final result will provide further insight into tournaments.

Combinatorial Aspects of Interval Orders and Interval Graphs