Polynomial Time For Graphs Research Articles

A recurrence for the surface area of (n, k)-arrangement graphs

An interesting property of an interconnected network is the number of nodes at distance i from an arbitrary processor, namely the node-centred surface area. This is an important property due to its applications in various fields of study. The ( n , k ) -arrangement graphs, denoted as A ( n , k ) are important class of graphs as they address the scalability issue in simpler topologies such as the hypercube and the star graph. Much work has been done to obtain a closed-form formula for the surface area of this class of graphs, but generally, it is not trivial to find an algorithm to compute the surface area of such graphs in polynomial time or to find an explicit formula with polynomially many terms in regard to the graph's parameters. In this paper, we present a simple recurrence that has linear computational complexity for A ( n , k ) for any arbitrary n and k in their defined range.

A complexity trichotomy for k-regular asymmetric spin systems with complex edge functions

We prove a complexity trichotomy theorem for a class of partition functions over k-regular graphs, where the signature is complex valued and not necessarily symmetric. In details, we establish explicit criteria, according to which the partition functions of all such systems are classified into three classes: For every parameter setting in C for the spin system, the partition function is either (1) computable in polynomial time for every graph, or (2) #P-hard for general graphs but computable in polynomial time for planar graphs, or (3) #P-hard even for planar 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.

Bisimplicial separators

AbstractA minimal separator of a graph is a set such that there exist vertices with the property that separates from in , but no proper subset of does. For an integer , we say that a minimal separator is ‐simplicial if it can be covered by cliques and denote by the class of all graphs in which each minimal separator is ‐simplicial. We show that for each , the class is closed under induced minors, and we use this to show that the Maximum Weight Stable Set problem can be solved in polynomial time for . We also give a complete list of minimal forbidden induced minors for . Next, we show that, for , every nonnull graph in has a ‐simplicial vertex, that is, a vertex whose neighborhood is a union of cliques; we deduce that the Maximum Weight Clique problem can be solved in polynomial time for graphs in . Further, we show that, for , it is NP‐hard to recognize graphs in ; the time complexity of recognizing graphs in is unknown. We also show that the Maximum Clique problem is NP‐hard for graphs in . Finally, we prove a decomposition theorem for diamond‐free graphs in (where the diamond is the graph obtained from by deleting one edge), and we use this theorem to obtain polynomial‐time algorithms for the Vertex Coloring and recognition problems for diamond‐free graphs in , and improved running times for the Maximum Weight Clique and Maximum Weight Stable Set problems for this class of graphs.

Computing the hull and interval numbers in the weakly toll convexity

A walk u0u1…uk−1uk of a graph G is a weakly toll walk if u0uk∉E(G), u0ui∈E(G) implies ui=u1, and ujuk∈E(G) implies uj=uk−1. The weakly toll interval of a set S⊆V(G), denoted by I(S), is formed by S and the vertices belonging to some weakly toll walk between two vertices of S. Set S is weakly toll convex if I(S)=S. The weakly toll convex hull of S, denote by H(S), is the minimum weakly toll convex set containing S. The weakly toll interval number of G is the minimum cardinality of a set S⊆V(G) such that I(S)=V(G); and the weakly toll hull number of G is the minimum cardinality of a set S⊆V(G) such that H(S)=V(G). In this work, we show how to compute the weakly toll interval and the weakly toll hull numbers of a graph in polynomial time. In contrast, we show that determining the weakly toll convexity number of a graph G (the size of a maximum weakly toll convex set different from V(G)) is NP-hard.

Finding dominating induced matchings in P10-free graphs in polynomial time

Let G=(V,E) be a finite undirected graph. An edge set E′⊆E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of E′. The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G; this problem is also known as the Efficient Edge Domination problem; it is the Efficient Domination problem for line graphs.The DIM problem is NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but is solvable in polynomial time for P9-free graphs [and in linear time for P7-free graphs] as well as for S1,2,4-free, for S2,2,2-free, and for S2,2,3-free graphs. In this paper, combining two distinct approaches, we solve it in polynomial time for P10-free graphs and introduce a partial result for the general case.

Relative timing information and orthology in evolutionary scenarios

BackgroundEvolutionary scenarios describing the evolution of a family of genes within a collection of species comprise the mapping of the vertices of a gene tree T to vertices and edges of a species tree S. The relative timing of the last common ancestors of two extant genes (leaves of T) and the last common ancestors of the two species (leaves of S) in which they reside is indicative of horizontal gene transfers (HGT) and ancient duplications. Orthologous gene pairs, on the other hand, require that their last common ancestors coincides with a corresponding speciation event. The relative timing information of gene and species divergences is captured by three colored graphs that have the extant genes as vertices and the species in which the genes are found as vertex colors: the equal-divergence-time (EDT) graph, the later-divergence-time (LDT) graph and the prior-divergence-time (PDT) graph, which together form an edge partition of the complete graph.ResultsHere we give a complete characterization in terms of informative and forbidden triples that can be read off the three graphs and provide a polynomial time algorithm for constructing an evolutionary scenario that explains the graphs, provided such a scenario exists. While both LDT and PDT graphs are cographs, this is not true for the EDT graph in general. We show that every EDT graph is perfect. While the information about LDT and PDT graphs is necessary to recognize EDT graphs in polynomial-time for general scenarios, this extra information can be dropped in the HGT-free case. However, recognition of EDT graphs without knowledge of putative LDT and PDT graphs is NP-complete for general scenarios. In contrast, PDT graphs can be recognized in polynomial-time. We finally connect the EDT graph to the alternative definitions of orthology that have been proposed for scenarios with horizontal gene transfer. With one exception, the corresponding graphs are shown to be colored cographs.

Intersection models and forbidden pattern characterizations 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.The complexity of recognizing 2-thin and proper 2-thin graphs is still open. In this work, we present characterizations of 2-thin and proper 2-thin graphs as intersection graphs of rectangles in the plane, as vertex intersection graphs of paths on a grid (VPG graphs), and by forbidden ordered patterns. We also prove that independent 2-thin graphs are exactly the interval bigraphs, and that proper independent 2-thin graphs are exactly the bipartite permutation graphs.Finally, we take a step towards placing the thinness and its variations in the landscape of width parameters, by upper bounding the proper thinness in terms of the bandwidth.

On the monophonic rank of a graph

A set of vertices $S$ of a graph $G$ is {\em monophonically convex} if every induced path joining two vertices of $S$ is contained in $S$. The {\em monophonic convex hull of $S$}, $\langle S \rangle$, is the smallest monophonically convex set containing $S$. A set $S$ is {\em monophonic convexly independent} if $v \not\in \langle S - \{v\} \rangle$ for every $v \in S$. The {\em monophonic rank} of $G$ is the size of the largest monophonic convexly independent set of $G$. We present a characterization of the monophonic convexly independent sets. Using this result, we show how to determine the monophonic rank of graph classes like bipartite, cactus, triangle-free, and line graphs in polynomial time. Furthermore, we show that this parameter can computed in polynomial time for $1$-starlike graphs, i.e., for split graphs, and that its determination is $\NP$-complete for $k$-starlike graphs for any fixed $k \ge 2$, a subclass of chordal graphs. We also consider this problem on the graphs whose intersection graph of the maximal prime subgraphs is a tree.

Algorithms and Complexity on Indexing Founder Graphs

We study the problem of matching a string in a labeled graph. Previous research has shown that unless the Orthogonal Vectors Hypothesis (OVH) is false, one cannot solve this problem in strongly sub-quadratic time, nor index the graph in polynomial time to answer queries efficiently (Equi et al. ICALP 2019, SOFSEM 2021). These conditional lower-bounds cover even deterministic graphs with binary alphabet, but there naturally exist also graph classes that are easy to index: For example, Wheeler graphs (Gagie et al. Theor. Comp. Sci. 2017) cover graphs admitting a Burrows-Wheeler transform -based indexing scheme. However, it is NP-complete to recognize if a graph is a Wheeler graph (Gibney, Thankachan, ESA 2019). We propose an approach to alleviate the construction bottleneck of Wheeler graphs. Rather than starting from an arbitrary graph, we study graphs induced from multiple sequence alignments (MSAs). Elastic degenerate strings (Bernadini et al. SPIRE 2017, ICALP 2019) can be seen as such graphs, and we introduce here their generalization: elastic founder graphs. We first prove that even such induced graphs are hard to index under OVH. Then we introduce two subclasses, repeat-free and semi-repeat-free graphs, that are easy to index. We give a linear time algorithm to construct a repeat-free (non-elastic) founder graph from a gapless MSA, and (parameterized) near-linear time algorithms to construct a semi-repeat-free (repeat-free, respectively) elastic founder graph from general MSA. Finally, we show that repeat-free founder graphs admit a reduction to Wheeler graphs in polynomial time.

On Proper Labellings of Graphs with Minimum Label Sum

The 1–2–3 Conjecture, raised by Karoński, Łuczak and Thomason, states that almost every graph G admits a proper 3-labelling, i.e., a labelling of the edges with 1, 2, 3 such that no two adjacent vertices are incident to the same sum of labels. Another interpretation of this conjecture, that may be attributed to Chartrand et al., is that almost every graph G can be turned into a locally irregular multigraph M, i.e., with no two adjacent vertices having the same degree, by replacing each of its edges by at most three parallel edges. In other words, for almost every graph G there should be a locally irregular multigraph M with the same adjacencies and having a relatively small number of edges. The 1–2–3 Conjecture, if true, would indeed imply that there is such an M with \(|E(M)| \le 3|E(G)|\). In this work, we study proper labellings of graphs with the extra requirement that the sum of assigned labels must be as small as possible. In other words, given a graph G, we are looking for a locally irregular multigraph \(M^*\) with the smallest number of edges possible that can be obtained from G by multiplying edges. This problem is actually quite different from the 1–2–3 Conjecture, as we prove that there is no absolute constant k such that \(M^*\) can always be obtained from G by replacing each edge with at most k parallel edges. We investigate several aspects of this problem, covering algorithmic and combinatorial aspects. In particular, we prove that the problem of designing proper labellings with minimum label sum is \({\mathcal {N}}{\mathcal {P}}\)-hard in general, but solvable in polynomial time for graphs with bounded treewidth. We also conjecture that for almost every connected graph G there should be a proper labelling with label sum at most 2|E(G)|, which we verify for several classes of graphs.

Complexity and algorithms for neighbor-sum-2-distinguishing {1,3}-edge-weighting of graphs

A neighbor-sum-distinguishing W-edge-weighting of a graph G=(V,E) is an assignment ω of weights from a set of integers W to the set of edges E of G such that for every pair of adjacent vertices, the incident sums induced by the edge-weighting are different, where the incident sum of a vertex v induced by the edge-weighting ω is σ(v)=∑u∈N(v)ω(uv), where N(v) is the set of neighbors of v in G. Whereas, a neighbor-sum-2-distinguishing W-edge-weighting of G is a neighbor-sum-distinguishing W-edge-weighting such that the incident sums of every pair of adjacent vertices differ by at least 2. A recently proposed conjecture about neighbor-sum-2-distinguishing {1,3,5}-edge-weighting states that any graph with no component isomorphic to K2 admits a neighbor-sum-2-distinguishing {1,3,5}-edge-weighting. In this paper, we prove that deciding whether there exists a neighbor-sum-2-distinguishing {1,3}-edge-weighting is NP-complete for bipartite graphs. We present an algorithm that computes a neighbor-sum-2-distinguishing {1,3}-edge-weighting of the central graph of any graph in polynomial time and thus proves the above conjecture for the central graph of any graph. Further, we establish that if any pair of graphs G and H admit neighbor-sum-2-distinguishing edge-weightings, then so does their Cartesian product, G□H. We study another aspect of neighbor-sum-2-distinguishing edge-weighting of a graph G, namely, the minimum number of incident sums used by a neighbor-sum-2-distinguishing {1,3}-edge-weighting, which is denoted by γΣ>1,{1,3}(G). We prove that the problems of deciding whether there exists a neighbor-sum-2-distinguishing {1,3}-edge-weighting of G from a given set of sums, and deciding whether γΣ>1,{1,3}(G)≤k are NP-complete for bipartite graphs. On the positive side, we provide both upper and lower bounds on γΣ>1,{1,3}(C(G)) for central graphs of bipartite graphs and split graphs. We also propose an algorithm that computes the optimal neighbor-sum-2-distinguishing {1,3}-edge-weighting of the central graphs of cycles and paths in polynomial time.

Finding Dominating Induced Matchings in $P_{10}$-Free Graphs in Polynomial Time

Let $G=(V,E)$ be a finite undirected graph. An edge subset $E' \subseteq E$ is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The \emph{Dominating Induced Matching} (\emph{DIM}) problem asks for the existence of a d.i.m. in $G$. The DIM problem is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but was solved in linear time for $P_7$-free graphs and in polynomial time for $P_8$-free graphs. In this paper, we solve it in polynomial time for $P_9$-free graphs.

Maximum weight independent sets for (S1,2,4,triangle)-free graphs in polynomial time

The Maximum Weight Independent Set (MWIS) problem on finite undirected graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum weight sum. MWIS is one of the most investigated and most important algorithmic graph problems; it is well known to be NP-complete, and it remains NP-complete even under various strong restrictions such as for triangle-free graphs. Its complexity for Pk-free graphs, k≥7, is an open problem. In [7], it is shown that MWIS can be solved in polynomial time for (P7,triangle)-free graphs. This result is extended by Maffray and Pastor [22] showing that MWIS can be solved in polynomial time for (P7,bull)-free graphs. In the same paper, they also showed that MWIS can be solved in polynomial time for (S1,2,3,bull)-free graphs.In this paper, using a similar approach as in [7], we show that MWIS can be solved in polynomial time for (S1,2,4,triangle)-free graphs which generalizes the result for (P7,triangle)-free graphs.

Thinness of product 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. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. In this paper we study the thinness and its variations of graph products. We show that the thinness behaves “well” in general for products, in the sense that for most of the graph products defined in the literature, the thinness of the product of two graphs is bounded by a function (typically product or sum) of their thinness, or of the thinness of one of them and the size of the other. We also show for some cases the non-existence of such a function.

Counting independent sets in graphs with bounded bipartite pathwidth

AbstractWe show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity , can be viewed as a strong generalization of Jerrum and Sinclair's work on approximately counting matchings, that is, independent sets in line graphs. The class of graphs with bounded bipartite pathwidth includes claw‐free graphs, which generalize line graphs. We consider two further generalizations of claw‐free graphs and prove that these classes have bounded bipartite pathwidth. We also show how to extend all our results to polynomially bounded vertex weights.

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.

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.

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.

Matching Cut in Graphs with Large Minimum Degree

In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be {mathsf {NP}}-complete. While Matching Cut is trivial for graphs with minimum degree at most one, it is {mathsf {NP}}-complete on graphs with minimum degree two. In this paper, we show that, for any given constant c>1, Matching Cut is {mathsf {NP}}-complete in the class of graphs with minimum degree c and this restriction of Matching Cut has no subexponential-time algorithm in the number of vertices unless the Exponential-Time Hypothesis fails. We also show that, for any given constant epsilon >0, Matching Cut remains {mathsf {NP}}-complete in the class of n-vertex (bipartite) graphs with unbounded minimum degree delta >n^{1-epsilon }. We give an exact branching algorithm to solve Matching Cut for graphs with minimum degree delta ge 3 in O^*(lambda ^n) time, where lambda is the positive root of the polynomial x^{delta +1}-x^{delta }-1. Despite the hardness results, this is a very fast exact exponential-time algorithm for Matching Cut on graphs with large minimum degree; for instance, the running time is O^*(1.0099^n) on graphs with minimum degree delta ge 469. Complementing our hardness results, we show that, for any two fixed constants 1< c <4 and c^{prime }ge 0, Matching Cut is solvable in polynomial time for graphs with large minimum degree delta ge frac{1}{c}n-c^{prime }.