Interval Graphs Research Articles

A heuristic for the minimum cost chromatic partition problem

The graph coloring problem consists in coloring the vertices of a graphG=(V, E)with a minimum number of colors, such as that any two adjacent vertices receive different colors. The minimum cost chromatic partition problem (MCCPP) is an extension of the graph coloring problem in which there are costs associated with the colors and one seeks a vertex coloring minimizing the sum of the costs of the colors used in each vertex. The problem finds applications in VLSI design and in some scheduling problems modeled on interval graphs. We propose a trajectory search heuristic using local search, path-relinking, and perturbations for solving MCCPP and discuss computational results.

Sortable Simplicial Complexes and $t$-Independence Ideals of Proper Interval Graphs

We introduce the notion of sortability and $t$-sortability for a simplicial complex and study the graphs for which their independence complexes are either sortable or $t$-sortable. We show that the proper interval graphs are precisely the graphs whose independence complex is sortable. By using this characterization, we show that the ideal generated by all squarefree monomials corresponding to independent sets of vertices of $G$ of size $t$ (for a given positive integer $t$) has the strong persistence property, when $G$ is a proper interval graph. Moreover, all of its powers have linear quotients.

On the Tractability of Optimization Problems on H-Graphs

For a graph H, a graph G is an H-graph if it is an intersection graph of connected subgraphs of some subdivision of H. H-graphs naturally generalize several important graph classes like interval graphs or circular-arc graph. This class was introduced in the early 1990s by Bíró, Hujter, and Tuza. Recently, Chaplick et al. initiated the algorithmic study of H-graphs by showing that a number of fundamental optimization problems like Maximum Clique, Maximum Independent Set, or Minimum Dominating Set are solvable in polynomial time on H-graphs. We extend and complement these algorithmic findings in several directions. First we show that for every fixed H, the class of H-graphs is of logarithmically-bounded boolean-width (via mim-width). Pipelined with the plethora of known algorithms on graphs of bounded boolean-width, this describes a large class of problems solvable in polynomial time on H-graphs. We also observe that H-graphs are graphs with polynomially many minimal separators. Combined with the work of Fomin, Todinca and Villanger on algorithmic properties of such classes of graphs, this identify another wide class of problems solvable in polynomial time on H-graphs. The most fundamental optimization problems among the problems solvable in polynomial time on H-graphs are Maximum Clique, Maximum Independent Set, and Minimum Dominating Set. We provide a more refined complexity analysis of these problems from the perspective of parameterized complexity. We show that Maximum Independent Set and Minimum Dominating Set are W[1]-hard being parameterized by the size of H plus the size of the solution. On the other hand, we prove that when H is a tree, then Minimum Dominating Set is fixed-parameter tractable parameterized by the size of H. For Maximum Clique we show that it admits a polynomial kernel parameterized by H and the solution size.

Dual Parameterization of Weighted Coloring

Given a graph G, a properk-coloring of G is a partition $$c = (S_i)_{i\in [1,k]}$$ of V(G) into k stable sets $$S_1,\ldots , S_{k}$$ . Given a weight function $$w: V(G) \rightarrow {\mathbb {R}}^+$$ , the weight of a color $$S_i$$ is defined as $$w(i) = \max _{v \in S_i} w(v)$$ and the weight of a coloringc as $$w(c) = \sum _{i=1}^{k}w(i)$$ . Guan and Zhu (Inf Process Lett 61(2):77–81, 1997) defined the weighted chromatic number of a pair (G, w), denoted by $$\sigma (G,w)$$ , as the minimum weight of a proper coloring of G. The problem of determining $$\sigma (G,w)$$ has received considerable attention during the last years, and has been proved to be notoriously hard: for instance, it is NP-hard on split graphs, unsolvable on n-vertex trees in time $$n^{o(\log n)}$$ unless the ETH fails, and W[1]-hard on forests parameterized by the size of a largest tree. We focus on the so-called dual parameterization of the problem: given a vertex-weighted graph (G, w) and an integer k, is $$\sigma (G,w) \le \sum _{v \in V(G)} w(v) - k$$ ? This parameterization has been recently considered by Escoffier (in: Proceedings of the 42nd international workshop on graph-theoretic concepts in computer science (WG). LNCS, vol 9941, pp 50–61, 2016), who provided an FPT algorithm running in time $$2^{{\mathcal {O}}(k \log k)} \cdot n^{{\mathcal {O}}(1)}$$ , and asked which kernel size can be achieved for the problem. We provide an FPT algorithm in time $$9^k \cdot n^{{\mathcal {O}}(1)}$$ , and prove that no algorithm in time $$2^{o(k)} \cdot n^{{\mathcal {O}}(1)}$$ exists under the ETH. On the other hand, we present a kernel with at most $$(2^{k-1}+1) (k-1)$$ vertices, and rule out the existence of polynomial kernels unless $$\mathsf{NP} \subseteq \mathsf{coNP} / \mathsf{poly}$$ , even on split graphs with only two different weights. Finally, we identify classes of graphs allowing for polynomial kernels, namely interval graphs, comparability graphs, and subclasses of circular-arc and split graphs, and in the latter case we present lower bounds on the degrees of the polynomials.

Min-Orderable Digraphs

We unify several seemingly different graph and digraph classes under one umbrella. These classes are all, broadly speaking, different generalizations of interval graphs, and include, in addition to interval graphs, adjusted interval digraphs, complements of threshold tolerance graphs (known as co-TT graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs. (The last three classes coincide, but have been investigated in different contexts.) We show that all of the above classes are united by a common ordering characterization, the existence of a min ordering. However, because the presence or absence of reflexive relationships (loops) affects whether a graph or digraph has a min ordering, to obtain this result, we must define the graphs and digraphs to have those loops that are implied by their definitions. These have been largely ignored in previous work. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, characterized by the existence of a compact representation, a signed-interval model, which is a generalization of known representations of the graph classes. We show that the signed-interval digraphs are precisely those digraphs that are characterized by the existence of a min ordering when the loops implied by the model are considered part of the graph. We also offer an alternative geometric characterization of these digraphs. We show that co-TT graphs are the symmetric signed-interval digraphs, the adjusted interval digraphs are the reflexive signed-interval digraphs, and the interval graphs are the intersection of these two classes, namely, the reflexive and symmetric signed-interval digraphs. Similar results hold for bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray bigraphs.

Research Slightly double fuzzy continuous functions via $e$-open sets P. Periyasamy 1 , A. Vadivel 2 *, G. Saravanakumar 3 and V. Chandrasekar 4 Malaya Journal of Matematik : Special Issue, Issue 1, 2020, Pages:570-575 DOI: 10.26637/MJM0S20/0109 | Full Article PDF | PDF size (1,412.53 KB) Research An $O(n)$-time algorithm to compute minimum $k$-hop connected dominating set of interval graphs

Research Slightly double fuzzy continuous functions via $e$-open sets P. Periyasamy 1 , A. Vadivel 2 *, G. Saravanakumar 3 and V. Chandrasekar 4 Malaya Journal of Matematik : Special Issue, Issue 1, 2020, Pages:570-575 DOI: 10.26637/MJM0S20/0109 | Full Article PDF | PDF size (1,412.53 KB) Research An $O(n)$-time algorithm to compute minimum $k$-hop connected dominating set of interval graphs

An $O(n)$-time algorithm to compute minimum $k$-hop connected dominating set of interval graphs

An $O(n)$-time algorithm to compute minimum $k$-hop connected dominating set of interval graphs

Optimum Frequency Allocation Mechanism of TV White Spaces by using Interval Graph

The unused frequencies of terrestrial TV are being explored in the recent years, to satisfy the bandwidth demands of ever-increasing wireless communication systems. Allocating these unused TV frequencies is often a challenging task. Allocations that fulfil the user requirements at various instants, while maximizing the utilization of available TV frequencies is the one that is desired. In this paper, interval-graph method is implemented to identify the optimum number of channels needed for the given demand of bandwidths. Simulations are carried out by using GLPK 4.65 solver. Results show the appropriate number of channels required or conversely the reduction of data rates to individual users based on the available bandwidths.

Optimum Frequency Allocation Mechanism of TV White Spaces by using Interval Graph

The unused frequencies of terrestrial TV are being explored in the recent years, to satisfy the bandwidth demands of ever-increasing wireless communication systems. Allocating these unused TV frequencies is often a challenging task. Allocations that fulfil the user requirements at various instants, while maximizing the utilization of available TV frequencies is the one that is desired. In this paper, interval-graph method is implemented to identify the optimum number of channels needed for the given demand of bandwidths. Simulations are carried out by using GLPK 4.65 solver. Results show the appropriate number of channels required or conversely the reduction of data rates to individual users based on the available bandwidths.

A linear algorithm for double Roman domination of proper interval graphs

A function [Formula: see text] is a double Roman dominating function on a graph [Formula: see text] if for every vertex [Formula: see text] with [Formula: see text] either there is a vertex [Formula: see text] with [Formula: see text] or there are distinct vertices [Formula: see text] with [Formula: see text] and for every vertex [Formula: see text] with [Formula: see text] there is a vertex [Formula: see text] with [Formula: see text]. The weight of a double Roman dominating function [Formula: see text] on [Formula: see text] is the value [Formula: see text]. The minimum weight of a double Roman dominating function on [Formula: see text] is called the double Roman domination number of [Formula: see text]. In this paper, we give an algorithm to compute the double Roman domination number of a given proper interval graph [Formula: see text] in [Formula: see text] time.

An Interval Graph with Alternate Cliiques of Size 3- Signed Roman Domination

An Interval Graph with Alternate Cliiques of Size 3- Signed Roman Domination

Coloring problem of signed interval graphs

A signed graph $(G,sigma)$ is a graph‎ ‎together with an assignment of signs ${+,-}$ to its edges where‎ ‎$sigma$ is the subset of its negative edges‎. ‎There are a few variants of coloring and clique problems of‎ ‎signed graphs‎, ‎which have been studied‎. ‎An initial version known as vertex coloring of signed graphs is defined by Zaslavsky in $1982$‎. ‎Recently Naserasr et. al., in [R‎. ‎Naserasr‎, ‎E‎. ‎Rollova and E‎. ‎Sopena‎, ‎Homomorphisms of signed graphs‎, ‎J‎. ‎Graph Theory‎, 79‎‎ (2015) 178--212, have defined signed chromatic and signed clique numbers of signed graphs‎. ‎In this paper we consider the latter mentioned problems for signed interval graphs‎. ‎We prove that the coloring problem of signed‎ ‎interval graphs is NP-complete whereas their ordinary coloring‎ ‎problem (the coloring problem of interval graphs) is in P‎. ‎Moreover we prove that the signed clique problem of a‎ ‎signed interval graph can be solved in polynomial time‎. ‎We also consider the‎ ‎complexity of further related problems‎.

The strong domination problem in block graphs and proper interval graphs

In a graph [Formula: see text], the degree of a vertex [Formula: see text], denoted by [Formula: see text], is defined as the number of edges incident on [Formula: see text]. A set [Formula: see text] of vertices of [Formula: see text] is called a strong dominating set if for every [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text]. For a given graph [Formula: see text], Min-Strong-DS is the problem of finding a strong dominating set of minimum cardinality. The decision version of Min-Strong-DS is shown to be NP -complete for chordal graphs. In this paper, we present polynomial time algorithms for computing a strong dominating set in block graphs and proper interval graphs, two subclasses of chordal graphs. On the other hand, we show that for a graph [Formula: see text] with [Formula: see text]-vertices, Min-Strong-DS cannot be approximated within a factor of [Formula: see text] for every [Formula: see text], unless NP [Formula: see text] DTIME ([Formula: see text]). We also show that Min-Strong-DS is APX -complete for graphs with maximum degree [Formula: see text]. On the positive side, we show that Min-Strong-DS can be approximated within a factor of [Formula: see text] for graphs with maximum degree [Formula: see text].

Removing Twins in Graphs to Break Symmetries

Determining vertex subsets are known tools to provide information about automorphism groups of graphs and, consequently about symmetries of graphs. In this paper, we provide both lower and upper bounds of the minimum size of such vertex subsets, called the determining number of the graph. These bounds, which are performed for arbitrary graphs, allow us to compute the determining number in two different graph families such are cographs and unit interval graphs.

Approximation algorithms for geometric conflict free covering problems

In the Geometric Conflict Free Covering, we are given a set of points P, a set of closed geometric objects O and a conflict graph CGO with O as vertex set. An edge (Oi,Oj) in CGO denotes conflict between Oi and Oj and at most one among these can be part of any feasible solution. A set of objects is conflict free if they form an independent set in CGO. The objective is to find a conflict free set of objects that maximizes the number of points covered.We consider the Unit Interval Covering where P is a set of points on the real line, and O is a set of closed unit-length intervals. The objective is to find a smallest subset of given intervals that covers P. We prove that for an arbitrary conflict graph the problem is Poly-APX -hard. We present an approximation algorithm for a special class of conflict graphs with a bounded graph parameter Clique Partition. A Clique Partition of the graph G is a set of cliques such that every vertex in the graph is part of exactly one clique. For any Clique Partition C, we define the Clique Partition Graph, GC with vertex set C and there is an edge (Ci,Cj) in GC, if and only if there exist two vertices in G, va∈Ci and vb∈Cj such that there is an edge (va,vb) in G. For a graph G, Clique Partition Chromatic Number is defined as the minimum chromatic number among all possible Clique Partitions of the Clique Partition Graph. In this paper, we consider those graph classes for which Clique Partition Chromatic Number can be computed in polynomial time.We present a 4γ approximation algorithm for conflict graphs having Clique Partition Chromatic Number γ. We show that unit interval graphs and unit disk graphs have constant Clique Partition Chromatic Number while for chordal graphs, it is bounded by log⁡n. Note that, Clique Partition Chromatic Number is less than or equal to the chromatic number. Thus our algorithm achieves a constant approximation factor for graphs with constant chromatic number (e.g. planar graphs ). This is the first result regarding the approximability in Geometric Conflict Free Covering.

Algorithmic aspects of upper paired-domination in graphs

A set D of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to a vertex in D and the subgraph induced by D contains a perfect matching (not necessarily as an induced subgraph). A paired-dominating set of G is minimal if no proper subset of it is a paired-dominating set of G. The upper paired-domination number of G, denoted by Γpr(G), is the maximum cardinality of a minimal paired-dominating set of G. In Upper-PDS, it is required to compute a minimal paired-dominating set with cardinality Γpr(G) for a given graph G. In this paper, we show that Upper-PDS cannot be approximated within a factor of n1−ε for any ε>0, unless P=NP and Upper-PDS is APX-complete for bipartite graphs of maximum degree 4. On the positive side, we show that Upper-PDS can be approximated within O(Δ)-factor for graphs with maximum degree Δ. We also show that Upper-PDS is solvable in polynomial time for threshold graphs, chain graphs, and proper interval graphs.

On the Carathéodory and exchange numbers of geodetic convexity in graphs

A set of vertices S of a graph G is geodesically convex if for every pair of vertices of S, all vertices of all shortest paths joining them also lie in S. The geodetic convex hull of S, denoted by 〈S〉, is the minimum geodesically convex set of G containing S. We say that S is Carathéodory independent if 〈S〉∖(⋃a∈S〈S∖{a}〉)≠∅ and that S is exchange independent if |S|=1 or there is p∈S such that 〈S∖{p}〉∖⋃a∈S∖{p}〈S∖{a}〉≠∅. The Carathéodory number (exchange number) of G is the cardinality of a maximum Carathéodory (exchange) independent set. In this paper, we show that deciding whether a given graph has exchange number at most k is an NP-complete problem. We also present characterizations of the Carathéodory and exchange numbers for some graph classes, like unit interval graphs and powers of cycles, which allow us to determine these parameters in polynomial time.

Algorithmic results on double Roman domination in graphs

Given a graph $$G=(V,E)$$, a function $$f:V\longrightarrow \{0,1,2,3\}$$ is called a double Roman dominating function on G if (i) for every $$v\in V$$ with $$f(v)=0$$, there are at least two neighbors of v that are assigned 2 under f or at least a neighbor of v that is assigned 3 under f, and (ii) for every vertex v with $$f(v)=1$$, there is at least one neighbor w of v with $$f(w)\ge 2$$. The weight of a double Roman dominating function f is $$f(V)=\sum _{u\in V}f(u)$$. The double Roman domination number of G, denoted by $$\gamma _{dR}(G)$$ is the minimum weight of a double Roman dominating function on G. For a graph $$G=(V,E)$$, Min-Double-RDF is to find a double Roman dominating function f with $$f(V)=\gamma _{dR}(G)$$. The decision version of Min-Double-RDF is shown to be NP-complete for chordal graphs and bipartite graphs. In this paper, we first strengthen the known NP-completeness of the decision version of Min-Double-RDF by showing that the decision version of Min-Double-RDF remains NP-complete for undirected path graphs, chordal bipartite graphs, and circle graphs. We then present linear time algorithms for computing the double Roman domination number in proper interval graphs and block graphs. We then discuss on the approximability of Min-Double-RDF and present a 2-approximation algorithm in 3-regular bipartite graphs.

Classes of Graphs with $e$-Positive Chromatic Symmetric Function

In the mid-1990s, Stanley and Stembridge conjectured that the chromatic symmetric functions of claw-free co-comparability (also called incomparability) graphs were $e$-positive. The quest for the proof of this conjecture has led to an examination of other, related graph classes. In 2013 Guay-Paquet proved that if unit interval graphs are $e$-positive, that implies claw-free incomparability graphs are as well. Inspired by this approach, we consider a related case and prove that unit interval graphs whose complement is also a unit interval graph are $e$-positive. We introduce the concept of strongly $e$-positive to denote a graph whose induced subgraphs are all $e$-positive, and conjecture that a graph is strongly $e$-positive if and only if it is (claw, net)-free.

Mim-width III. Graph powers and generalized distance domination problems

We generalize the family of (σ,ρ) problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as Distance-rDominating Set and Distance-rIndependent Set. We show that these distance problems are in XP parameterized by the structural parameter mim-width, and hence polynomial-time solvable on graph classes where mim-width is bounded and quickly computable, such as k-trapezoid graphs, Dilworth k-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, k-polygon graphs, circular arc graphs, complements of d-degenerate graphs, and H-graphs if given an H-representation. We obtain these results by showing that taking any power of a graph never increases its mim-width by more than a factor of two. To supplement these findings, we show that many classes of (σ,ρ) problems are W[1]-hard parameterized by mim-width + solution size.We show that powers of graphs of tree-width w−1 or path-width w and powers of graphs of clique-width w have mim-width at most w. These results provide new classes of bounded mim-width. We prove a slight strengthening of the first statement which implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most 1.