Permutation Graphs Research Articles

On the Weisfeiler–Leman Dimension of Permutation Graphs

On the Weisfeiler–Leman Dimension of Permutation Graphs

Counting kernels in directed graphs with arbitrary orientations

A kernel of a directed graph is a subset of vertices that is both independent and absorbing (every vertex not in the kernel has an out-neighbour in the kernel).Not all directed graphs contain kernels, and computing a kernel or deciding that none exist is NP-complete even on low-degree planar digraphs. The existing polynomial-time algorithms for this problem are all obtained by restricting both the undirected structure and the edge orientations of the input: for example, to chordal graphs without bidirectional edges (Pass-Lanneau, Igarashi and Meunier, Discrete Appl Math 2020) or to permutation graphs where each clique has a sink (Abbas and Saoula, 4OR 2005).By contrast, we count the kernels of a fuzzy circular interval graph in polynomial time, regardless of its edge orientations, and return a kernel when one exists. (Fuzzy circular interval graphs were introduced by Chudnovsky and Seymour in their structure theorem for claw-free graphs.)We also consider kernels on cographs, where we establish NP-hardness in general but linear-time solvability on the subclass of threshold graphs.

Partial and Simultaneous Transitive Orientations via Modular Decompositions

A natural generalization of the recognition problem for a geometric graph class is the problem of extending a representation of a subgraph to a representation of the whole graph. A related problem is to find representations for multiple input graphs that coincide on subgraphs shared by the input graphs. A common restriction is the sunflower case where the shared graph is the same for each pair of input graphs. These problems translate to the setting of comparability graphs where the representations correspond to transitive orientations of their edges. We use modular decompositions to improve the runtime for the orientation extension problem and the sunflower orientation problem to linear time. We apply these results to improve the runtime for the partial representation problem and the sunflower case of the simultaneous representation problem for permutation graphs to linear time. We also give the first efficient algorithms for these problems on circular permutation graphs.

Partitioning subclasses of chordal graphs with few deletions

In the (Vertex)k-Way Cut problem, input is an undirected graph G, an integer s, and the goal is to find a subset S of edges (vertices) of size at most s, such that G−S has at least k connected components. Downey et al. [Electr. Notes Theor. Comput. Sci. 2003] showed that k-Way Cut is W[1]-hard parameterized by k. However, Kawarabayashi and Thorup [FOCS 2011] showed that the problem is fixed-parameter tractable (FPT) in general graphs with respect to the parameter s and provided a O(ssO(s)n2) time algorithm, where n denotes the number of vertices in G. The best-known algorithm for this problem runs in time sO(s)nO(1) given by Lokshtanov et al. [ACM Tran. of Algo. 2021]. On the other hand, Vertexk-Way Cut is W[1]-hard with respect to either of the parameters, k or s or k+s. These algorithmic results motivate us to look at the problems on special classes of graphs. In this paper, we consider the (Vertex)k-Way Cut problem on subclasses of chordal graphs and obtain the following results.•We first give a sub-exponential FPT algorithm for k-Way Cut running in time 2O(slog⁡s)nO(1) on chordal graphs.•It is “known” that Vertexk-Way Cut is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k+s. We complement this hardness result by designing polynomial-time algorithms for Vertexk-Way Cut on interval graphs, circular-arc graphs and permutation graphs.

Resolving prime modules: The structure of pseudo-cographs and galled-tree explainable graphs

The modular decomposition of a graph G is a natural construction to capture key features of G in terms of a labeled tree (T,t) whose vertices are labeled as “series” (1), “parallel” (0) or “prime”. However, full information of G is provided by its modular decomposition tree (T,t) only, if G is a cograph, i.e., G does not contain prime modules. In this case, (T,t) explains G, i.e., {x,y}∈E(G) if and only if the lowest common ancestor lcaT(x,y) of x and y has label “1”. Pseudo-cographs, or, more generally, GaTEx graphs G are graphs that can be explained by labeled galled-trees, i.e., labeled networks (N,t) that are obtained from the modular decomposition tree (T,t) of G by replacing the prime vertices in T by simple labeled cycles. GaTEx graphs can be recognized and labeled galled-trees that explain these graphs can be constructed in linear time.In this contribution, we provide a novel characterization of GaTEx graphs in terms of a set FGT of 25 forbidden induced subgraphs. This characterization, in turn, allows us to show that GATEX graphs are closely related to many other well-known graph classes such as P4-sparse and P4-reducible graphs, weakly-chordal graphs, perfect graphs with perfect order, comparability and permutation graphs, murky graphs as well as interval graphs, Meyniel graphs or very strongly-perfect and brittle graphs. Moreover, we show that every GATEX graph as twin-width at most 1.

A class of graphs with large rankwidth

We describe several graphs with arbitrarily large rankwidth (or equivalently with arbitrarily large cliquewidth). Korpelainen, Lozin, and Mayhill [Split permutation graphs, Graphs and Combinatorics, 30(3):633–646, 2014] proved that there exist split graphs with Dilworth number 2 with arbitrarily large rankwidth, but without explicitly constructing them. We provide an explicit construction. Maffray, Penev, and Vušković [Coloring rings, Journal of Graph Theory 96(4):642-683, 2021] proved that graphs that they call rings on n sets can be colored in polynomial time. We show that for every fixed integer n≥3, there exist rings on n sets with arbitrarily large rankwidth. When n≥5 and n is odd, this provides a new construction of even-hole-free graphs with arbitrarily large rankwidth.

Hereditary classes of ordered sets of width at most two

This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs.Several open problems about hereditary classes of relational structures which have been considered over the years have a positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and Mayhill, respectively by Albert, Brignall, Ruškuc and Vatter.In this paper we present an overview of properties of these hereditary classes in the framework of the Theory of Relations as presented by Roland Fraïssé.We provide another proof of the results mentioned above. It is based on the existence of a countable universal poset of width two, obtained by the first author in 1978, his notion of multichainability (1978) (a kind of analog to letter-graphs), and metric properties of incomparability graphs. Using Laver’s theorem (1971) on better-quasi-ordering (bqo) of countable chains we prove that a wqo hereditary class of finite or countable bipartite permutation graphs is necessarily bqo. This gives a positive answer to a conjecture of Nash-Williams (1965) in this case. We extend a previous result of Albert et al. by proving that if a hereditary class of finite, respectively countable, bipartite permutation graphs is wqo, respectively bqo, then the corresponding hereditary classes of posets of width at most two and bichains are wqo, respectively bqo.Several notions of labelled wqo are also considered. We prove that they are all equivalent in the case of bipartite permutation graphs, posets of width at most two and the corresponding bichains. We characterize hereditary classes of finite bipartite permutation graphs which remain wqo when labels from a wqo are added. Hereditary classes of posets of width two, bipartite permutation graphs and the corresponding bichains having finitely many bounds are also characterized.We prove that a hereditary class of finite bipartite permutation graphs is not wqo if and only if it embeds the poset of finite subsets of N ordered by set inclusion. This answers a long standing conjecture of the first author in the case of bipartite permutation graphs.

Path eccentricity of graphs

Let G be a connected graph. The eccentricity of a path P, denoted by eccG(P), is the maximum distance from P to any vertex in G. In the Central path (CP) problem, our aim is to find a path of minimum eccentricity. This problem was introduced by Cockayne et al., in 1981, in the study of different centrality measures on graphs. They showed that CP can be solved in linear time in trees, but it is known to be NP-hard in many classes of graphs such as chordal bipartite graphs, planar 3-connected graphs, split graphs, etc. We investigate the path eccentricity of a connected graph G as a parameter. Let pe(G) denote the value of eccG(P) for a central path P of G. We obtain tight upper bounds for pe(G) in some graph classes. We show that pe(G)≤1 on biconvex graphs and design an algorithm that finds such a path in linear time. On the other hand, by investigating the longest paths of a graph, we obtain tight upper bounds for pe(G) on arbitrary graphs and k-connected graphs. Finally, we study the relation between a central path and a longest path in a graph. We show that, on bipartite permutation graphs, a longest path is also a central path. Furthermore, for any such class of graphs, we exhibit a superclass that does not satisfy this property.

Permutation graphs with co-inversion on Γ<sub>1</sub> - non-deranged permutations

In this paper, we define permutation graphs on Γ1 -non-deranged permutations using the set of co-inversion as edge set, and the values of permutation as the set of vertices. From the graphs, we observed that diameter and radius of the graph of any ω1 is one, the graph of any ωp-1 ∈ GΓ1p simple, the graph of ω1 is completed and other properties of the graphs were also observed.

Critical properties of bipartite permutation graphs

AbstractThe class of bipartite permutation graphs enjoys many nice and important properties. In particular, this class is critically important in the study of clique‐ and rank‐width of graphs, because it is one of the minimal hereditary classes of graphs of unbounded clique‐ and rank‐width. It also contains a number of important subclasses, which are critical with respect to other parameters, such as graph lettericity or shrub‐depth, and with respect to other notions, such as well‐quasi‐ordering or complexity of algorithmic problems. In the present paper we identify critical subclasses of bipartite permutation graphs of various types.

Locally irregular edge-coloring of subcubic graphs

A graph is locally irregular if no two adjacent vertices have the same degree. A locally irregular edge-coloring of a graph G is such an (improper) edge-coloring that the edges of any fixed color induce a locally irregular subgraph. Among the graphs admitting a locally irregular edge-coloring, i.e., decomposable graphs, only one is known to require 4 colors, while for all the others it is believed that 3 colors suffice. In this paper, we prove that decomposable claw-free graphs with maximum degree 3, all cycle permutation graphs, and all generalized Petersen graphs admit a locally irregular edge-coloring with at most 3 colors. We also discuss when 2 colors suffice for a locally irregular edge-coloring of cubic graphs and present an infinite family of cubic graphs of girth 4 which require 3 colors.

Graph universal cycles: Compression and connections to universal cycles

Universal cycles, such as De Bruijn cycles, are cyclic sequences of symbols that represent every combinatorial object from some family exactly once as a consecutive subsequence. Graph universal cycles are a graph analogue of universal cycles introduced in 2010. We introduce graph universal partial cycles, a more compact representation of graph classes, which use “do not know” edges. We show how to construct graph universal partial cycles for labeled graphs, threshold graphs, and permutation graphs. For threshold graphs and permutation graphs, we demonstrate that the graph universal cycles and graph universal partial cycles are closely related to universal cycles and compressed universal cycles, respectively. Using the same connection, for permutation graphs, we define and prove the existence of an s-overlap form of graph universal cycles. We also prove the existence of a generalized form of graph universal cycles for unlabeled graphs.

Strong watermark numbers encoded as reducible permutation graphs against edge modification attacks

Software watermarking is a defense technique used to prevent or discourage software piracy by embedding a signature in the code. In (Discrete Applied Mathematics 250 (2018) 145–164), a software watermarking system is presented which encodes an integer number w (i.e., a watermark) as a reducible permutation flow-graph F [ π ∗ ] embeddable in the code through the use of a self-inverting permutation π ∗ . In this work, we theoretically investigate this watermarking system and exploit structural properties of the self-inverting permutation π ∗ encoding the watermark in order to prove its resilience to edge-modification attacks on the flow-graph F [ π ∗ ]. Based on the minimum number of edge modifications needed to be applied on F [ π ∗ ] so that a different watermark can be extracted from the resulting graph, we give a characterization of the watermarks as strong, intermediate or weak and provide good recommendations for the choices of watermark.

MaxCut on permutation graphs is NP‐complete

AbstractThe decision problem MaxC ut is known to be NP‐complete since the seventies, but only recently its restriction to interval graphs has been announced to be hard by Adhikary, Bose, Mukherjee, and Roy. Building on their proof, in this paper we prove that the M axC ut problem is NP‐complete on permutation graphs. This settles a long‐standing open problem that appeared in the 1985 column of the Ongoing Guide to NP‐completeness by David S. Johnson, and is the first NP‐hardness entry for permutation graphs in such column.

Testing independence of bivariate censored data using random walk on restricted permutation graph

In this paper, we propose a procedure to test the independence of bivariate censored data, which is generic and applicable to any censoring types in the literature. To test the hypothesis, we consider a rank-based statistic, Kendall’s tau statistic. The censored data defines a restricted permutation space of all possible ranks of the observations. We propose the statistic, the average of Kendall’s tau over the ranks in the restricted permutation space. To evaluate the statistic and its reference distribution, we develop a Markov chain Monte Carlo (MCMC) procedure to obtain uniform samples on the restricted permutation space and numerically approximate the null distribution of the averaged Kendall’s tau. We numerically compare the power of our procedure to existing state of the art procedures in the literature under various censoring types. We apply the procedure to three real data examples with different censoring types, and compare the results with those by existing methods.

On finding separators in temporal split and permutation graphs

We study the NP-hard Temporal(s,z)-Separation problem on temporal graphs, which are graphs with fixed vertex sets but edge sets that change over discrete time steps. We tackle Temporal(s,z)-Separation by restricting the layers (i.e., graphs characterized by edges that are present at a certain point in time) to specific graph classes. We restrict the layers of the temporal graphs to be either all split graphs or all permutation graphs (both being perfect graph classes) and provide both intractability and tractability results. In particular, we show that in general Temporal(s,z)-Separation remains NP-hard both on temporal split and temporal permutation graphs, but we also spot promising islands of fixed-parameter tractability particularly based on parameterizations that measure the amount of “change over time”.

Permutation graphs and the weak Bruhat order

Permutation graphs are graphs associated with permutations where edges represent inversions. We study different classes of permutation graphs and  isomorphic permutation graphs. A complete answer of a basic isomorphism question is given for trees. A connection to the majorization order for integer partitions is established, in connection with the notion of inversion vectors. Finally, we study special transitivity properties related to inversions of permutations.

Succinct Permutation Graphs

We present a succinct data structure for permutation graphs, and their superclass of circular permutation graphs, i.e., data structures using optimal space up to lower order terms. Unlike concurrent work on circle graphs (Acan et al. in Theor Comput Sci, https://doi.org/10.1016/j.tcs.2022.06.022, 2022), our data structure also supports distance and shortest-path queries, as well as adjacency and neighborhood queries, all in optimal time. We present in particular the first succinct exact distance oracle for (circular) permutation graphs. A second succinct data structure also supports degree queries in time independent of the neighborhood’s size at the expense of an O(log n/log log n)-factor overhead in all running times. Furthermore, we develop a succinct data structure for the class of bipartite permutation graphs. We demonstrate how to run algorithms directly over our succinct representations for several problems on permutation graphs: Clique, Coloring, Independent Set, Hamiltonian Cycle, All-Pair Shortest Paths, and others. Finally, we initiate the study of semi-distributed graph representations; a concept that smoothly interpolates between distributed (labeling schemes) and centralized (standard data structures). We show how to turn some of our data structures into semi-distributed representations by storing only O(n) bits of additional global information, circumventing the lower bound on distance labeling schemes for permutation graphs.

Algorithms for maximum internal spanning tree problem for some graph classes

For a given graph G, a maximum internal spanning tree of G is a spanning tree of G with maximum number of internal vertices. The Maximum Internal Spanning Tree (MIST) problem is to find a maximum internal spanning tree of the given graph. The MIST problem is a generalization of the Hamiltonian path problem. Since the Hamiltonian path problem is NP-hard, even for bipartite and chordal graphs, two important subclasses of graphs, the MIST problem also remains NP-hard for these graph classes. In this paper, we propose linear-time algorithms to compute a maximum internal spanning tree of cographs, block graphs, cactus graphs, chain graphs and bipartite permutation graphs. The optimal path cover problem, which asks to find a path cover of the given graph with maximum number of edges, is also a well studied problem. In this paper, we also study the relationship between the number of internal vertices in maximum internal spanning tree and number of edges in optimal path cover for the special graph classes mentioned above.

Graph Bipartization Problem with Applications to Via Minimization in VLSI Design

The bipartization problem for a graph [Formula: see text] asks for finding a subset [Formula: see text] of [Formula: see text] such that the induced subgraph [Formula: see text] is bipartite and [Formula: see text] is maximized. This problem has significant applications in the via minimization of VLSI design. The problem has been proved NP-complete and the fixed parameter solvability has been known in the literature. This paper presents several polynomial-time algorithms for special graph families, such as split graphs, co-bipartite graphs, chordal graphs, and permutation graphs.