scholarly journals An efficientg-centroid location algorithm for cographs

2005 ◽  
Vol 2005 (9) ◽  
pp. 1405-1413 ◽  
Author(s):  
V. Prakash
Keyword(s):  
Location Problem ◽  
Maximum Clique ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Outerplanar Graphs ◽  
Arbitrary Graph ◽  
Maximal Outerplanar Graphs ◽  
Split Graphs ◽  
Location Algorithm ◽  
Centroid Location

In 1998, Pandu Rangan et al. Proved that locating theg-centroid for an arbitrary graph is𝒩𝒫-hard by reducing the problem of finding the maximum clique size of a graph to theg-centroid location problem. They have also given an efficient polynomial time algorithm for locating theg-centroid for maximal outerplanar graphs, Ptolemaic graphs, and split graphs. In this paper, we present anO(nm)time algorithm for locating theg-centroid for cographs, wherenis the number of vertices andmis the number of edges of the graph.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

scholarly journals Some New Classes of Open Distance-Pattern Uniform Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Bibin K. Jose
Keyword(s):  
Nonempty Subset ◽  
Chordal Graphs ◽  
Outerplanar Graph ◽  
Outerplanar Graphs ◽  
Induced Subgraphs ◽  
Minimum Cardinality ◽  
Maximal Outerplanar Graphs ◽  
Split Graphs ◽  
Strongly Chordal Graphs

Given an arbitrary nonempty subset M of vertices in a graph G=(V,E), each vertex u in G is associated with the set fMo(u)={d(u,v):v∈M,u≠v} and called its open M-distance-pattern. The graph G is called open distance-pattern uniform (odpu-) graph if there exists a subset M of V(G) such that fMo(u)=fMo(v) for all u,v∈V(G), and M is called an open distance-pattern uniform (odpu-) set of G. The minimum cardinality of an odpu-set in G, if it exists, is called the odpu-number of G and is denoted by od(G). Given some property P, we establish characterization of odpu-graph with property P. In this paper, we characterize odpu-chordal graphs, and thereby characterize interval graphs, split graphs, strongly chordal graphs, maximal outerplanar graphs, and ptolemaic graphs that are odpu-graphs. We also characterize odpu-self-complementary graphs, odpu-distance-hereditary graphs, and odpu-cographs. We prove that the odpu-number of cographs is even and establish that any graph G can be embedded into a self-complementary odpu-graph H, such that G and G¯ are induced subgraphs of H. We also prove that the odpu-number of a maximal outerplanar graph is either 2 or 5.

scholarly journals Uniquely monopolar-partitionable block graphs

2014 ◽  
Vol Vol. 16 no. 2 (PRIMA 2013) ◽  
Author(s):  
Xuegang Chen ◽  
Jing Huang
Keyword(s):  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Special Issue ◽  
Block Graphs ◽  
International Audience ◽  
Split Graphs ◽  
Np Complete ◽  
Vertex Partitions ◽  
General Graphs

Special issue PRIMA 2013 International audience As a common generalization of bipartite and split graphs, monopolar graphs are defined in terms of the existence of certain vertex partitions. It has been shown that to determine whether a graph has such a partition is NP-complete for general graphs and polynomial for several classes of graphs. In this paper, we investigate graphs that admit a unique such partition and call them uniquely monopolar-partitionable graphs. By employing a tree trimming technique, we obtain a characterization of uniquely monopolar-partitionable block graphs. Our characterization implies a polynomial time algorithm for recognizing them.

scholarly journals Recognizing generalized Sierpiński graphs

2020 ◽  
Vol 14 (1) ◽  
pp. 122-137
Author(s):  
Wilfried Imrich ◽  
Iztok Peterin
Keyword(s):  
Polynomial Time ◽  
Complete Graph ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Arbitrary Graph ◽  
Base Graph ◽  
Sierpiński Graphs ◽  
Vertex Set

Let H be an arbitrary graph with vertex set V (H) = [nH] = {l,?, nH}. The generalized Sierpi?ski graph SnH , n ? N, is defined on the vertex set [nH]n, two different vertices u = un ?u1 and v = vn ? v1 being adjacent if there exists an h? [n] such that (a) ut = vt, for t > h, (b) uh ? vh and uhvh ? E(H), and (c) ut = vh and vt = uh for t < h. If H is the complete graph Kk, then we speak of the Sierpi?ski graph Sn k . We present an algorithm that recognizes Sierpi?ski graphs Sn k in O(|V (Sn k )|1+1=n) = O(|E(Sn k )|) time. For generalized Sierpi?ski graphs SnH we present a polynomial time algorithm for the case when H belong to a certain well defined class of graphs. We also describe how to derive the base graph H from an arbitrarily given SnH .

A General Model and Efficient Algorithms for Reliable Facility Location Problem Under Uncertain Disruptions

2021 ◽  
Author(s):  
Yongzhen Li ◽  
Xueping Li ◽  
Jia Shu ◽  
Miao Song ◽  
Kaike Zhang
Keyword(s):  
Facility Location ◽  
Location Problem ◽  
Facility Location Problem ◽  
Cutting Plane ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Cutting Plane Algorithm ◽  
Robust Model ◽  
Stochastic Case ◽  
Distributionally Robust

This paper studies the reliable uncapacitated facility location problem in which facilities are subject to uncertain disruptions. A two-stage distributionally robust model is formulated, which optimizes the facility location decisions so as to minimize the fixed facility location cost and the expected transportation cost of serving customers under the worst-case disruption distribution. The model is formulated in a general form, where the uncertain joint distribution of disruptions is partially characterized and is allowed to have any prespecified dependency structure. This model extends several related models in the literature, including the stochastic one with explicitly given disruption distribution and the robust one with moment information on disruptions. An efficient cutting plane algorithm is proposed to solve this model, where the separation problem is solved respectively by a polynomial-time algorithm in the stochastic case and by a column generation approach in the robust case. Extensive numerical study shows that the proposed cutting plane algorithm not only outperforms the best-known algorithm in the literature for the stochastic problem under independent disruptions but also efficiently solves the robust problem under correlated disruptions. The practical performance of the robust models is verified in a simulation based on historical typhoon data in China. The numerical results further indicate that the robust model with even a small amount of information on disruption correlation can mitigate the conservativeness and improve the location decision significantly. Summary of Contribution: In this paper, we study the reliable uncapacitated facility location problem under uncertain facility disruptions. The problem is formulated as a two-stage distributionally robust model, which generalizes several related models in the literature, including the stochastic one with explicitly given disruption distribution and the robust one with moment information on disruptions. To solve this generalized model, we propose a cutting plane algorithm, where the separation problem is solved respectively by a polynomial-time algorithm in the stochastic case and by a column generation approach in the robust case. The efficiency and effectiveness of the proposed algorithm are validated through extensive numerical experiments. We also conduct a data-driven simulation based on historical typhoon data in China to verify the practical performance of the proposed robust model. The numerical results further reveal insights into the value of information on disruption correlation in improving the robust location decisions.

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