Interval Graphs Research Articles

Reconstruction and verification of chordal graphs with a distance oracle

A hidden graph is a graph whose edge set is hidden. A distance oracle of a graph G is a black-box that receives two vertices of G and outputs the distance between the two vertices. Given a hidden graph, the reconstruction problem aims to identify the edges of the hidden graph by accessing a distance oracle, and the verification problem aims to check whether the hidden graph is equal to another given graph (not hidden).If the hidden graph G is a connected chordal graph, a Las Vegas reconstruction algorithm using O(Δ32Δ⋅n(2Δ+log2⁡n)log⁡n) distance queries is known, where Δ is the maximum degree of G and n is the number of vertices of G. Improving upon this result, we present a reconstruction algorithm using only O(Δ2nlog2⁡n) distance queries. As a byproduct, we obtain a deterministic algorithm for the verification of chordal graphs with O(Δ2nlog⁡n) distance queries. Additionally, we derive a deterministic algorithm of reconstructing connected interval graphs using only O(Δn) distance queries, and prove that reconstructing or verifying a connected interval graph needs Ω(Δn) distance queries, which implies that this algorithm is the best possible in terms of the number of distance queries needed.

A sufficient condition for the unpaired k-disjoint path coverability of interval graphs

Given disjoint source and sink sets, $$S = \{ s_1,\ldots ,s_k\}$$ and $$T = \{t_1,\ldots , t_k\}$$ , in a graph G, an unpaired k-disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths $$\{ P_1, \ldots , P_k\}$$ that altogether cover every vertex of the graph, in which $$P_i$$ is a path from source $$s_i$$ to some sink $$t_j$$ . In this paper, we prove that if the scattering number, $${{\,\mathrm{sc}\,}}(G)$$ , of an interval graph G of order $$n \ge 2k$$ is less than or equal to $$-k$$ , there exists an unpaired k-disjoint path cover joining S and T in G for any possible configurations of source and sink sets S and T of size k each. The bound $${{\,\mathrm{sc}\,}}(G) \le -k$$ is tight; moreover, the proof directly leads to a quadratic algorithm for building an unpaired k-disjoint path cover.

Graphs with Two Moplexes

Moplexes are natural graph structures that arise when lifting Dirac’s classical theorem from chordal graphs to general graphs. The notion is known to be closely related to lexicographic searches in graphs as well as to asteroidal triples, and has been applied in several algorithms related to graph classes such as interval graphs, claw-free, and diamond-free graphs. However, while every non-complete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is, among others, motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are interval.In this work we initiate an investigation of k-moplex graphs, which are defined as graphs containing at most k moplexes. Of particular interest is the smallest nontrivial case, k = 2, which forms a counterpart to the class of interval graphs. As our main structural result, we show that the class of connected 2-moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely Graph Isomorphism and Max-Cut. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets. We also discuss sufficient conditions for the existence of Hamiltonian paths in 2-moplex graphs as well as connections with avoidable vertices.

$L(p,q)$-labeling of graphs with interval representations

We provide upper bounds on the $L(p,q)$-labeling number of graphs which have interval (or circular-arc) representations via simple greedy algorithms. We prove that there exists an $L(p,q)$-labeling with span at most $\max\{2(p+q-1)\Delta-4q+2, (2p-1)\mu+(2q-1)\Delta-2q+1\}$ for interval $k$-graphs, $\max\{p,q\}\Delta$ for interval graphs, $3\max\{p,q\}\Delta+p$ for circular-arc graphs, $2(p+q-1)\Delta-2q+1$ for permutation graphs and $(2p-1)\Delta+(2q-1)(\mu-1)$ for cointerval graphs. In particular, these improve existing bounds on $L(p,q)$-labeling of interval graphs and $L(2,1)$-labeling of permutation graphs. Furthermore, we provide upper bounds on the coloring of the squares of aforementioned classes.

The chromatic index of split-interval graphs

The chromatic index of a graph G, χ'(G), is the least number of colors of some edge coloring of G such that no two adjacent edges have the same color. Given a graph G and an integer k, to decide if χ'(G) ≤ k is NP-complete. A graph is an interval graph if it represents the intersection relation of a set of closed intervals in R. A graph is a split graph if its vertex set can be partitioned into a clique and an independent set. A graph is split-interval if it is simultaneously an interval and a split graph. In this paper we show how to determine the chromatic index of all split-interval graphs. Our proof leads to a polynomial-time algorithm to deciding if χ'(G) ≤ k given an integer k and a split-interval graph G.

Intersection models for 2-thin and proper 2-thin graphs

The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Graphs with thinness at most two include, for example, bipartite convex graphs. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. Proper thinness is defined analogously, generalizing proper interval graphs, and a larger family of NP-complete problems are known to be polynomially solvable for graphs with bounded proper thinness. It is known that the thinness of a graph is at most its pathwidth plus one. In this work, we prove that the proper thinness of a graph is at most its bandwidth, for graphs with at least one edge. It is also known that boxicity is a lower bound for the thinness. The main results of this work are characterizations of 2-thin and 2-proper thin graphs as intersection graphs of rectangles in the plane with sides parallel to the Cartesian axes and other specific conditions. We also bound the bend number of graphs with low thinness as vertex intersection graphs of paths on a grid (Bk-VPG graphs are the graphs that have a representation in which each path has at most k bends). We show that 2-thin graphs are a subclass of B1-VPG graphs (moreover, of L-graphs, that is, B1-VPG graphs admitting a representation that uses only one of the four possible shapes ⌊,⌋⌈,⌉), and that 3-thin graphs are a subclass of B3-VPG graphs. We also show that B0-VPG graphs may have arbitrarily large thinness, and that not every 4-thin graph is a VPG graph.

Parameterized algorithms for locating-dominating sets

A locating-dominating set D of a graph G is a dominating set of G where each vertex not in D has a unique neighborhood in D, and the Locating-Dominating Set problem asks if G contains such a dominating set of bounded size. This problem is known to be NP-hard even on restricted graph classes, such as interval graphs, split graphs, and planar bipartite subcubic graphs. On the other hand, it is known to be solvable in polynomial time for some graph classes, such as trees and, more generally, graphs of bounded cliquewidth. While these results have numerous implications on the parameterized complexity of the problem, little is known in terms of kernelization under structural parameterizations. In this work, we begin filling this gap in the literature. Our first result shows that Locating-Dominating Set is W[1]-hard when parameterized by the size of a minimum clique cover. We present an exponential kernel for the distance to cluster parameterization and show that, unless NP ⊆ coNP/poly, no polynomial kernel exists for Locating-Dominating Set when parameterized by vertex cover nor when parameterized by distance to clique. We then turn our attention to parameters not bounded by either of the previous two, and exhibit a linear kernel when parameterizing by the max leaf number; in this context, we leave the parameterization by feedback edge set as the primary open problem in our study.

Algorithmic complexity of outer independent Roman domination and outer independent total Roman domination

Let $$G=(V,E)$$ be a graph. A function $$f : V \rightarrow \{0, 1, 2\}$$ is an outer independent Roman dominating function (OIRDF) on a graph G if for every vertex $$v \in V$$ with $$f (v) = 0$$ there is a vertex u adjacent to v with $$f (u) = 2$$ and $$\{x\in V:f(x)=0\}$$ is an independent set. The weight of f is the value $$ f(V)=\sum _{v\in V}f(v)$$ . An outer independent total Roman dominating function (OITRDF) f on G is an OIRDF on G such that for every $$v\in V$$ with $$f(v)>0$$ there is a vertex u adjacent to v with $$f (u)>0$$ . The minimum weight of an OIRDF on G is called the outer independent Roman domination number of G, denoted by $$\gamma _{oiR}(G)$$ . Similarly, the outer independent total Roman domination number of G is defined, denoted by $$\gamma _{oitR}(G)$$ . In this paper, we first show that computing $$\gamma _{oiR}(G)$$ (respectively, $$\gamma _{oitR}(G)$$ ) is a NP-hard problem, even when G is a chordal graph. Then, for a given proper interval graph $$G=(V,E)$$ we propose an algorithm to compute $$\gamma _{oiR}(G)$$ (respectively, $$\gamma _{oitR}(G)$$ ) in $${\mathcal {O}}(|V| )$$ time.

Computing the weighted neighbor isolated tenacity of interval graphs in polynomial time

AbstractWeighted graphs in graph theory are created by weighing different values depending on the importance of connections or centers in a graph model. Networks can be modeled with graphs such that the devices and centers correspond to the vertices and connections correspond to the edges. In these networks, weight can be assigned to the vertices for the workload and importance of the devices and centers, so that planning such as security and cost can be made in advance in the design of the network. Network reliability and security is an important issue in the computing area. There are several parameters for vulnerability measurement values of these networks modeled with graphs. We recommend the weighted conversion of the neighbor isolated tenacity parameter for this topic. It is known that tenacity, which is the basis of this parameter, is NP‐hard. But polynomial solutions can be created in interval graphs, which is a special graph from the perfect graph class. In this article, polynomial time algorithm is given to calculate weighted neighbor isolated tenacity of the interval graphs.

Injective coloring of some subclasses of bipartite graphs and chordal graphs

A vertex coloring of a graph G=(V,E) that uses k colors is called an injectivek-coloring of G if no two vertices having a common neighbor have the same color. The minimum k for which G has an injectivek-coloring is called the injective chromatic number of G. Given a graph G and a positive integer k, the Decide Injective Coloring Problem is to decide whether G admits an injective k-coloring. It is known that Decide Injective Coloring Problem is NP-complete for bipartite graphs. In this paper, we strengthen this result by showing that this problem remains NP-complete for perfect elimination bipartite graphs, star-convex bipartite graphs and comb-convex bipartite graphs, which are proper subclasses of bipartite graphs. Moreover, we show that for every ϵ>0, it is not possible to efficiently approximate the injective chromatic number of a perfect elimination bipartite graph within a factor of n13−ϵ unless ZPP = NP. On the positive side, we propose a linear time algorithm for biconvex bipartite graphs and O(nm) time algorithm for convex bipartite graphs for finding the optimal injective coloring. We prove that the injective chromatic number of a chordal bipartite graph can be determined in polynomial time. It is known that Decide Injective Coloring Problem is NP-complete for chordal graphs. We give a linear time algorithm for computing the injective chromatic number of proper interval graphs, which is a proper subclass of chordal graphs. Decide Injective Coloring Problem is also known to be NP-complete for split graphs. We show that Decide Injective Coloring Problem remains NP-complete for K1,t-free split graphs for t≥4 and polynomially solvable for t≤3.

Hardness and algorithms of equitable tree-coloring problem in chordal graphs

An equitable tree-k-coloring of a graph is a vertex k-coloring such that each color class induces a forest and the size of any two color classes differs by at most one. In this work, we show that every interval graph G has an equitable tree-k-coloring for any integer k≥⌈(Δ(G)+1)/2⌉, solving a conjecture of Wu, Zhang and Li (2013) for interval graphs, and furthermore, give a linear-time algorithm for determining whether a proper interval graph admits an equitable tree-k-coloring for a given integer k. For disjoint union of split graphs, or K1,r-free interval graphs with r≥4, we prove that it is W[1]-hard to decide whether there is an equitable tree-k-coloring when parameterized by number of colors, or by treewidth, number of colors and maximum degree, respectively. On the positive side, we propose a quadratic 2-approximation algorithm for the equitable tree-coloring problem in chordal graphs. Moreover, it is proved that there is no α-approximation algorithm for any α<32.

CPG graphs: Some structural and hardness results

In this paper we continue the systematic study of Contact graphs of Paths on a Grid (CPG graphs) initiated in Deniz et al. (2018). A CPG graph is a graph for which there exists a collection of pairwise interiorly disjoint paths on a grid in one-to-one correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. If every such path has at most k bends for some k≥0, the graph is said to be Bk-CPG.We first show that, for any k≥0, the class of Bk-CPG graphs is strictly contained in the class of Bk+1-CPG graphs even within the class of planar graphs, thus implying that there exists no k≥0 such that every planar CPG graph is Bk-CPG. The main result of the paper is that recognizing CPG graphs and Bk-CPG graphs with k≥1 is NP-complete. Moreover, we show that the same remains true even within the class of planar graphs in the case k≥3. We then consider several graph problems restricted to CPG graphs and show, in particular, that Independent Set and Clique Cover remain NP-hard for B0-CPG graphs. Finally, we consider the related classes Bk-EPG of edge-intersection graphs of paths with at most k bends on a grid. Although it is possible to optimally color a B0-EPG graph in polynomial time, as this class coincides with that of interval graphs, we show that, in contrast, 3-Colorability is NP-complete for B1-EPG graphs.

A fixed-parameter algorithm for scheduling unit dependent tasks on parallel machines with time windows

This paper proves that the existence of a feasible schedule for a set of dependent tasks of unit execution times with release dates and deadlines on a limited number of processors is a fixed-parameter tractable problem. The parameter considered is the pathwidth of the interval graph associated with the time windows of the tasks. A fixed-parameter algorithm based on a dynamic programming approach is developed and proved to solve this decision problem.Fixed-parameter algorithms for the two classical problems P|prec,pi=1,ri|Cmax and P|prec,pi=1,ri|Lmax are then derived using a binary search. They are, as far as we know, the first fixed-parameter algorithms for these scheduling problems.

An $$O(n+m)$$ time algorithm for computing a minimum semitotal dominating set in an interval graph

Let $$G=(V,E)$$ be a graph without isolated vertices. A set $$D\subseteq V$$ is said to be a dominating set of G if for every vertex $$v\in V\setminus D$$ , there exists a vertex $$u\in D$$ such that $$uv\in E$$ . A set $$D\subseteq V$$ is called a semitotal dominating set of G if D is a dominating set and every vertex in D is within distance 2 from another vertex of D. For a given graph G, the semitotal domination problem is to find a semitotal dominating set of G with minimum cardinality. The decision version of the semitotal domination problem is shown to be NP-complete for chordal graphs and bipartite graphs. Henning and Pandey (Theor Comput Sci 766:46–57, 2019) proposed an $$O(n^2)$$ time algorithm for computing a minimum semitotal dominating set in interval graphs. In this paper, we show that for a given interval graph $$G=(V,E)$$ , a minimum semitotal dominating set of G can be computed in $$O(n+m)$$ time, where $$n=|V|$$ and $$m=|E|$$ . This improves the complexity of the semitotal domination problem for interval graphs from $$O(n^2)$$ to $$O(n+m)$$ .

Approaching the One-Sided Exemplar Adjacency Number Problem.

The one-sided Exemplar Adjacency Number (EAN) is a known problem for computing the exemplar similarity between a generic linear genome G with gene duplications and an exemplar genome H (over the same set of n gene families). In this problem, we need to compute an exemplar genome G, which is a permutation obtained from G, such that the number of common adjacencies between G and H is maximized. Unfortunately, the problem is not only NP-hard but also NP-hard to approximate. In this paper, we approach the problem by relaxing the constraint such that a sub-permutation G+ obtained from G does not have to include all the gene families, but still needs to have a length at least k. Hence G+ is called a pseudo-exemplar genome. Then, a slightly more general problem (One-sided EAN+) is defined: compute a pseudo-exemplar genome G+ from G such that the number of common adjacencies between H and G+ is maximized. Certainly One-sided EAN+ contains One-sided EAN as a special case; moreover, it presents some flexibility in designing algorithms. First, we relax and formulate the One-sided EAN+ problem as the maximum independent set (MIS) on a colored interval graph and hence reduce the appearance of each gene to at most two times. We show that this new relaxation is still NP-complete, though a simple factor-2 approximation algorithm can be designed; moreover, we also prove that the problem cannot be approximated within 2-ε by a local search technique. We then show that this relaxed version is fixed-parameter tractable (FPT). Second, to ensure that each gene appears in G+ at most once, we use integer linear programming (ILP) to solve this problem. Finally, we implement our algorithm and compare it with the up-to-date software GREDU, with simulated signed and unsigned genomes. It turns out that our algorithm is more stable and can process genomes of length up to 12,000 for signed genomes (while GREDU can falter on such a large signed genome and it cannot handle unsigned genomes at all).

Total Roman domination for proper interval graphs

A function f : V → {0,1,2} is a total Roman dominating function (TRDF) on a graph G =( V,E ) if for every vertex v ∈ V with f ( v ) = 0 there is a vertex u adjacent to v with f ( u ) = 2 and for every vertex v ∈ V with f ( v ) &gt; 0 there exists a vertex u ∈ N G ( v ) with f ( u ) &gt; 0. The weight of a total Roman dominating function f on G is equal to f ( V )=Σ v ∈ V f ( v ). The minimum weight of a total Roman dominating function on G is called the total Roman domination number of G . In this paper, we give an algorithm to compute the total Roman domination number of a given proper interval graph G =( V,E ) in O (| V |) time.

Interval graph multi-coloring-based resource reservation for energy-efficient containerized cloud data centers

Containerized deployment of microservices has quickly become a well-known virtualization technology due to its higher portability, scalability, good isolation, and lightweight solutions. However, it faces several challenges in terms of the capital and operational expenses in large-scale data centers. In particular, services in cloud are usually instantiated as a group of containers, which continuously trigger frequent communication workloads and hence significantly degrades the service performance due to inefficient allocation of containers. Thus to deploy microservices, service providers must consider different types of objectives, such as optimizing the communication cost or the operational cost, which are joint objectives that have previously been studied independently. In this paper, we study the problem of communication-aware container-based advance reservation to optimize the energy and communication cost for microservices deployment. We applied the interval graph model to map the container reservation scenario of microservices and derived various performance bounds. Then, we propose greedy graph multi-coloring-based centralized and distributed algorithms to find an efficient allocation of containers. Through theoretical analysis and extensive experimental results, we demonstrate that the proposed approaches can decrease the total cost by up to 31% compared to the current state-of-the-art methods.

Interval graph of facial regions with common intersection salient points for identifying and classifying facial expression

Measuring the facial expression is an important research and used in many real-time applications. Various methods are proposed in the academia and industry for a decade and still continue to have research potential. This paper proposes a novel scheme by using Interval graph of facial regions. It is assumed that common intersecting salient points of facial regions can be used for estimating the emotions. The facial region is decomposed in four sub regions and the Interval graph is extracted for each region. The common salient points and degree of deformation and direction of deformation are measured for vertical, horizontal and diagonal directions. These values are considered as feature vectors. The well-known datasets such as JAFEE and CK++ are used for evaluating the performance of various classification algorithms and estimating their average classification accuracy. The average classification of the proposed approach is 95.9% and 94.7% for CK++ dataset and JAFEE dataset respectively. The performance of the proposed approach is better when compared to other state of art approaches.

Spiders and their Kin: An Investigation of Stanley’s Chromatic Symmetric Function for Spiders and Related Graphs

We study the chromatic symmetric functions of graph classes related to spiders, namely generalized spider graphs (line graphs of spiders), and what we call horseshoe crab graphs. We show that no two generalized spiders have the same chromatic symmetric function, thereby extending the work of Martin, Morin and Wagner. Additionally, we establish Query ID=Q3 Text=Please confirm if the inserted city name in affiliation [3] is correct. Amend if necessary. that a subclass of generalized spiders, which we call generalized nets, has no e-positive members, providing a more general counterexample to the necessity of the claw-free condition. We use yet another class of generalized spiders to construct a counterexample to a problem involving the e-positivity of claw-free, $$P_4$$ -sparse graphs, showing that Tsujie’s result on the e-positivity of claw-free, $$P_4$$ -free graphs cannot be extended to graphs in this set. Finally, we investigate the e-positivity of another type of graphs, the horseshoe crab graphs (a class of unit interval graphs), and prove the positivity of all but one of the coefficients. This has close connections to the work of Gebhard and Sagan and Cho and Huh.

Unit Ball Graphs on Geodesic Spaces

Consider finitely many points in a geodesic space. If the distance of two points is less than a fixed threshold, then we regard these two points as "near". Connecting near points with edges, we obtain a simple graph on the points, which is called a unit ball graph. If the space is the real line, then it is known as a unit interval graph. Unit ball graphs on a geodesic space describe geometric characteristics of the space in terms of graphs. In this article, we show that every unit ball graph on a geodesic space is (strongly) chordal if and only if the space is an $ \mathbb{R} $-tree and that every unit ball graph on a geodesic space is (claw, net)-free if and only if the space is a connected manifold of dimension at most $ 1 $. As a corollary, we prove that the collection of unit ball graphs essentially characterizes the real line and the unit circle.