Graph realizations: Maximum degree in vertex neighborhoods

본 논문은 NP-완전 문제인 간선 색칠과 그래프 부류 결정 문제를 동시에 해결하는 O(E)의 다항시간 알고리즘을 제안하였다. 제안된 알고리즘은 최대차수-최소차수 정점 쌍 간선을 단순히 선택하는 방법으로 간선 채색수 <TEX>${\chi}^{\prime}(G)$</TEX>를 결정하였다. 결정된 <TEX>${\chi}^{\prime}(G)$</TEX>는 <TEX>${\Delta}(G)$</TEX> 또는 <TEX>${\Delta}(G)+1$</TEX>을 얻는다. 결국, 알고리즘 수행 결과 얻은 <TEX>${\chi}^{\prime}(G)$</TEX>로부터 <TEX>${\chi}^{\prime}(G)={\Delta}(G)$</TEX>이면 부류 1, <TEX>${\chi}^{\prime}(G)={\Delta}(G)+1$</TEX>이면 부류 2로 분류할 수 있다. 또한, 미해결 문제로 알려진 "최대차수가 6인 단순, 평면 그래프는 부류 1이다."라는 Vizing의 평면 그래프 추정도 증명하였다. This paper proposes a O(E) polynomial-time algorithm that has been devised to simultaneously solve edge-coloring problem and graph classification problem both of which remain NP-complete. The proposed algorithm selects an edge connecting maximum and minimum degree vertices so as to determine the number of edge coloring <TEX>${\chi}^{\prime}(G)$</TEX>. Determined <TEX>${\chi}^{\prime}(G)$</TEX> is in turn either <TEX>${\Delta}(G)$</TEX> or <TEX>${\Delta}(G)+1$</TEX>. Eventually, the result could be classified as class 1 if <TEX>${\chi}^{\prime}(G)={\Delta}(G)$</TEX> and as category 2 if <TEX>${\chi}^{\pri</TEX><TEX>me}(G)={\Delta}(G)+1$</TEX>. This paper also proves Vizing's planar graph conjecture, which states that 'all simple, planar graphs with maximum degree six or seven are of class one, closing the remaining possible case', which has known to be NP-complete.

Given a vertex-weighted tree T, the split of an edge xy in T is min{s_x(xy), s_y(xy)} where s_u(uv) is the sum of all weights of vertices that are closer to u than to v in T. Given a set of weighted vertices V and a multiset of splits S, we consider the problem of constructing a tree on V whose splits correspond to S. The problem is known to be NP-complete, even when all vertices have unit weight and the maximum vertex degree of T is required to be no more than 4. We show that the problem is strongly NP-complete when T is required to be a path, the problem is NP-complete when all vertices have unit weight and the maximum degree of T is required to be no more than 3, and it remains NP-complete when all vertices have unit weight and T is required to be a caterpillar with unbounded hair length and maximum degree at most 3. We also design polynomial time algorithms for the variant where T is required to be a path and the number of distinct vertex weights is constant, and the variant where all vertices have unit weight and T has a constant number of leaves. The latter algorithm is not only polynomial when the number of leaves, k, is a constant, but also fixed-parameter tractable when parameterized by k. Finally, we shortly discuss the problem when the vertex weights are not given but can be freely chosen by an algorithm. The considered problem is related to building libraries of chemical compounds used for drug design and discovery. In these inverse problems, the goal is to generate chemical compounds having desired structural properties, as there is a strong correlation between structural properties, such as the Wiener index, which is closely connected to the considered problem, and biological activity.

Graphical games have been proposed as a game-theoretic model of large-scale distributed networks of non-cooperative agents. When the number of players is large, and the underlying graph has low degree, they provide a concise way to represent the players' payoffs. It has recently been shown that the problem of finding Nash equilibria in a general degree-3 graphical game with two actions per player is complete for the complexity class PPAD, indicating that it is unlikely that there is any polynomial-time algorithm for this problem. In this paper, we study the complexity of graphical games with two actions per player on bounded-degree trees. This setting was first considered by Kearns, Littman and Singh, who proposed a dynamic programming-based algorithm that computes all Nash equilibria of such games. The running time of their algorithm is exponential, though approximate equilibria can be computed efficiently. Later, Littman, Kearns and Singh proposed a modification to this algorithm that can find a single Nash equilibrium in polynomial time. We show that this modified algorithm is incorrect-the output is not always a Nash equilibrium. We then propose a new algorithm that is based on the ideas of Kearns et al. and computes all Nash equilibria in quadratic time if the input graph is a path, and in polynomial time if it is an arbitrary graph of maximum degree 2. Moreover, our algorithm can be used to compute Nash equilibria of graphical games on arbitrary trees, but the running time can be exponential, even when the tree has bounded degree. We show that this is inevitable -- any algorithm of this type will take exponential time, even on bounded-degree trees with pathwidth 2. It is an open question whether our algorithm runs in polynomial time on graphs with pathwidth 1, but we show that finding a Nash equilibrium for a 2-action graphical game in which the underlying graph has maximum degree 3 and constant pathwidth is PPAD-complete (so is unlikely to be tractable).

Understanding spatial correlation is vital in many fields including epidemiology and social science. Lee, Meeks and Pettersson (Stat. Comput. 2021) recently demonstrated that improved inference for areal unit count data can be achieved by carrying out modifications to a graph representing spatial correlations; specifically, they delete edges of the planar graph derived from border-sharing between geographic regions in order to maximise a specific objective function. In this paper we address the computational complexity of the associated graph optimisation problem. We demonstrate that this problem cannot be solved in polynomial time unless P = NP; we further show intractability for two simpler variants of the problem. We follow these results with two parameterised algorithms that exactly solve the problem in polynomial time in restricted settings. The first of these utilises dynamic programming on a tree decomposition, and runs in polynomial time if both the treewidth and maximum degree are bounded. The second algorithm is restricted to problem instances with maximum degree three, as may arise from triangulations of planar surfaces, but is an FPT algorithm when the maximum number of edges that can be removed is taken as the parameter.

Realizability of p-point graphs with prescribed minimum degree, maximum degree, and point-connectivity

A vertex subset I of a graph G is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from I. The k -Path Vertex Cover Reconfiguration (k -PVCR) problem asks if one can transform one k-path vertex cover into another via a sequence of k-path vertex covers where each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. We investigate the computational complexity of k -PVCR from the viewpoint of graph classes under the well-known reconfiguration rules: \(\mathsf {TS}\), \(\mathsf {TJ}\), and \(\mathsf {TAR}\). The problem for \(k=2\), known as the Vertex Cover Reconfiguration (VCR) problem, has been well-studied in the literature. We show that certain known hardness results for VCR on different graph classes including planar graphs, bounded bandwidth graphs, chordal graphs, and bipartite graphs, can be extended for k -PVCR. In particular, we prove a complexity dichotomy for k -PVCR on general graphs: on those whose maximum degree is 3 (and even planar), the problem is \(\mathtt {PSPACE}\)-complete, while on those whose maximum degree is 2 (i.e., paths and cycles), the problem can be solved in polynomial time. Additionally, we also design polynomial-time algorithms for k -PVCR on trees under each of \(\mathsf {TJ}\) and \(\mathsf {TAR}\). Moreover, on paths, cycles, and trees, we describe how one can construct a reconfiguration sequence between two given k-path vertex covers in a yes-instance. In particular, on paths, our constructed reconfiguration sequence is shortest.

The Complexity of the Matching-cut Problem for Various Graph Classes

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2013 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Totally odd subdivisions and parity subdivisions: Structures and ColoringKen-ichi KawarabayashiKen-ichi Kawarabayashipp.1013 - 1029Chapter DOI:https://doi.org/10.1137/1.9781611973105.73PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract A totally odd H-subdivision means a subdivision of a graph H in which each edge of H corresponds to a path of odd length. Thus this concept is a generalization of a subdivision of H. In this paper, we give a structure theorem for graphs without a fixed graph H as a totally odd subdivision. Namely, every graph with no totally odd H-subdivision has a tree-decomposition such that each piece is either 1. after deleting bounded number of vertices, an “almost” embedded graph into a bounded-genus surface, or 2. after deleting bounded number of vertices, a bipartite graph, or 3. after deleting bounded number of vertices, a graph with maximum degree at most f(|H|) for some function f of |H| (or a 6|H|-degenerate graph). Moreover, we can obtain either a totally odd Kk-subdivision or such a tree-decomposition in polynomial time. We note that for minor-free graphs, we just need the first structure [37], while for odd-minor-free graphs, we need the first two structures [10]. For subdivision-free graphs, we need the first and the third structures [17, 29]. So our result can be viewed as a combination of odd-minor-free graphs and subdivision-free graphs. The same conclusion of the structure theorem is true if we replace “totally odd” by “parity”. Hence this generalizes the structure theorem for subdivision-free graphs [17, 29]. We also consider coloring of graphs with no totally odd Kk-subdivision. We prove that any graph with no totally odd Kk-subdivision is 79k2/4-colorable. The bound on the chromatic number is essentially best possible since the correct order of the magnitude of the chromatic number even for graphs with no Kk-subdivision is Θ(k2). Our result improves the bound given by Thomassen [44]. Furthermore, it also generalizes the result by Bollobás and Thomason, and Komlós and Szemerédi, [6, 30] for graphs without Kk-subdivisions. Finally we consider of coloring graphs with no totally odd Kk-subdivision in terms of an algorithmic view. Using our structure theorem, we give an approximation algorithm for coloring of a graph G without a fixed graph H as a totally odd subdivision, using 2χ(G) + 6(|H| − 1) colors, where χ(G) is chromatic number of G. The same conclusion is true if we replace “totally odd” by “parity”. We point out that it is Unique-Game hard to obtain an O(k/ log2 k)-approximation algorithm for graph-coloring of graphs with maximum degree at most k − 2 [2], and hence it is also Unique-Game hard to obtain an O(k/ log2 k)-approximation algorithm for graph-coloring of graphs even without a Kk-subdivision (i.e, without the parity constraint). Thus our additive error Θ(k) is most likely best possible up to constant. Previous chapter Next chapter RelatedDetails Published:2013ISBN:978-1-61197-251-1eISBN:978-1-61197-310-5 https://doi.org/10.1137/1.9781611973105Book Series Name:ProceedingsBook Code:PR143Book Pages:xix + 1915Key words:Totally odd subdivision, structure theorem, coloring, polynomial time

We consider the problem of finding a local optimum for the Max-Cut problem with FLIP-neighborhood, in which exactly one node changes the partition. Schaffer and Yannakakis (SICOMP, 1991) showed PLS-completeness of this problem on graphs with unbounded degree. On the other side, Poljak (SICOMP, 1995) showed that in cubic graphs every FLIP local search takes O(n2) steps, where n is the number of nodes. Due to the huge gap between degree three and unbounded degree, Ackermann, Roglin, and Vocking (JACM, 2008) asked for the smallest d such that on graphs with maximum degree d the local MAX-CUT problem with FLIP-neighborhood is PLS-complete. In this paper, we prove that the computation of a local optimum on graphs with maximum degree five is PLS-complete. Thus, we solve the problem posed by Ackermann et al. almost completely by showing that d is either four or five (unless PLS ⊆ P). On the other side, we also prove that on graphs with degree O(log n) every FLIP local search has probably polynomial smoothed complexity. Roughly speaking, for any instance, in which the edge weights are perturbated by a (Gaussian) random noise with variance &sigma2, every FLIP local search terminates in time polynomial in n and σ-1, with probability 1-n-Ω(1). Putting both results together, we may conclude that although local MAX-CUT is likely to be hard on graphs with bounded degree, it can be solved in polynomial time for slightly perturbated instances with high probability.

A spanning subgraph $S = ( V,E^\prime )$ of a connected simple graph $G = ( V,E )$ is a $f( x )$-spanner if for any pair of nodes u and $v $, $d_S ( u,v ) \leq f ( d_G ( u,v ) )$ where $d_G $ and $d_S $ are the usual distance functions in graphs G and S, respectively. The delay of the $f( x )$-spanner is $f( x ) - x$. In this paper $( 2.5\sqrt {( 3x + 6 )/4} + 6 + x )$-spanners for two-dimensional grids with maximum degree 3 are found and it is proven that the delay of these spanners is within a constant factor of optimal. A $( \frac{1}{k}x + k + 8 - \frac{7}{k} + x )$-spanner of the X-tree with maximum degree 3 is described, and it is proven that the delay of this spanner is within a constant factor of optimal. In addition, a $( 2 + x )$-spanner of the pyramid with maximum degree 6 and a $( \frac{1}{k}x + k + 8 - \frac{7}{k} + x )$-spanner of the pyramid with maximum degree 5 are described, and it is proven that the delay of the latter spanner is within a constant factor of optimal.

This paper discusses the complexity of packingk-chains (simple paths of lengthk) into an undirected graph; the chains packed must be either vertex-disjoint or edge-disjoint. Linear-time algorithms are given for both problems when the graph is a tree, and for the edge-disjoint packing problem when the graph is general andk = 2. The vertex-disjoint packing problem for general graphs is shown to be NP-complete even when the graph has maximum degree three andk = 2. Similarly the edge-disjoint packing problem is NP-complete even when the graph has maximum degree four andk = 3.

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. Let d be a fixed positive integer. We show that there is a natural number N(d) such that if T is any tree with m ≥ N(d) edges and maximum degree at most d, then the harmonious chromatic number h(T) is k or k + 1, where k is the least positive integer such that . We also give a polynomial time algorithm for determining the harmonious chromatic number of a tree with maximum degree at most d.

Article Free Access Share on Exact sampling and approximate counting techniques Author: Mark Huber School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NYView Profile Authors Info & Claims STOC '98: Proceedings of the thirtieth annual ACM symposium on Theory of computingMay 1998 Pages 31–40https://doi.org/10.1145/276698.276709Published:23 May 1998Publication History 25citation666DownloadsMetricsTotal Citations25Total Downloads666Last 12 Months66Last 6 weeks15 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

A clique of a graph G is a set of pairwise adjacent vertices of G. A clique-coloring of G is an assignment of colors to the vertices of G in such a way that no inclusion-wise maximal clique of size at least two of G is monochromatic. An equitable clique-coloring of G is a clique-coloring such that any two color classes differ in size by at most one. Bacso and Tuza proved that connected claw-free graphs with maximum degree at most four, other than chordless odd cycles of order greater than three, are 2-clique-colorable and a 2-clique-coloring can be found in $$O(n^{2})$$ Bacso and Tuza (Discrete Math Theor Comput Sci 11(2):15–24, 2009). In this paper we prove that every connected claw-free graph with maximum degree at most four, not a chordless odd cycle of order greater than three, has an equitable 2-clique-coloring. In addition we improve the algorithm described in the paper mentioned by giving an equitable 2-clique-coloring in linear time for this class of graphs.

We present an algorithm for testing the k-vertex-connectivity of graphs with given maximum degree. The time complexity of the algorithm is independent of the number of vertices and edges of graphs. A graph G with n vertices and maximum degree at most d is called ε-far from k-vertex-connectivity when at least \(\frac{\epsilon dn}{2}\) edges must be added to or removed from G to obtain a k-vertex-connected graph with maximum degree at most d. The algorithm always accepts every graph that is k-vertex-connected and rejects every graph that is ε-far from k-vertex-connectivity with a probability of at least 2/3. The algorithm runs in \({O\left(d\left(\frac{c}{\epsilon d}\right)^{k}\log\frac{1}{\epsilon d}\right)}\) time (c > 1 is a constant) for given (k − 1)-vertex-connected graphs, and \({O\left(d\left(\frac{ck}{\epsilon d}\right)^{k}\log\frac{k}{\epsilon d}\right)}\) time (c > 1 is a constant) for given general graphs. It is the first constant-time k-vertex-connectivity testing algorithm for general k ≥ 4.KeywordsMaximum DegreeGeneral GraphTesting AlgorithmAdjacency ListVertex ConnectivityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.