scholarly journals An O(mn^2) algorithm for computing the strong geodetic number in outerplanar graphs

2020 ◽  
Author(s):  
Mauro Mezzini
Keyword(s):  
Outerplanar Graphs ◽  
Geodetic Number ◽  
Strong Geodetic Number

scholarly journals Strong Geodetic Number in Some Networks

2019 ◽  
Vol 11 (2) ◽  
pp. 20
Author(s):  
Huifen Ge ◽  
Zhao Wang ◽  
Jinyu Zou
Keyword(s):  
Upper Bound ◽  
Cartesian Product ◽  
Geodetic Number ◽  
Product Graphs ◽  
Geodetic Set ◽  
Cartesian Product Graphs ◽  
Strong Geodetic Number

A vertex subset S of a graph is called a strong geodetic set if there exists a choice of exactly one geodesic for each pair of vertices of S in such a way that these (|S| 2) geodesics cover all the vertices of graph G. The strong geodetic number of G, denoted by sg(G), is the smallest cardinality of a strong geodetic set. In this paper, we give an upper bound of strong geodetic number of the Cartesian product graphs and study this parameter for some Cartesian product networks.

Strong Geodetic Number of Graphs and Connectivity

2019 ◽  
Vol 43 (3) ◽  
pp. 2443-2453
Author(s):  
Zhao Wang ◽  
Yaping Mao ◽  
Huifen Ge ◽  
Colton Magnant
Keyword(s):  
Geodetic Number ◽  
Strong Geodetic Number

scholarly journals Strong geodetic problem on Cartesian products of graphs

2018 ◽  
Vol 52 (1) ◽  
pp. 205-216 ◽  
Author(s):  
Vesna Iršič ◽  
Sandi Klavžar
Keyword(s):  
Shortest Path ◽  
Upper Bound ◽  
Cartesian Product ◽  
Cartesian Products ◽  
Cartesian Product Of Graphs ◽  
Geodetic Number ◽  
Recent Variation ◽  
Product Of Graphs ◽  
Strong Geodetic Number ◽  
Cartesian Products Of Graphs

The strong geodetic problem is a recent variation of the geodetic problem. For a graph G, its strong geodetic number sg(G) is the cardinality of a smallest vertex subset S, such that each vertex of G lies on a fixed shortest path between a pair of vertices from S. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for sg(G □ H) is determined, as well as exact values for Km □ Kn, K1,k □ Pl, and prisms over Kn–e. Connections between the strong geodetic number of a graph and its subgraphs are also discussed.

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 Algorithmic upper bounds for graph geodetic number

2021 ◽  
Author(s):  
Ahmad T. Anaqreh ◽  
Boglárka G.-Tóth ◽  
Tamás Vinkó
Keyword(s):  
Upper Bounds ◽  
Geodetic Number

scholarly journals Minimum rank of outerplanar graphs

2012 ◽  
Vol 436 (9) ◽  
pp. 3701-3720
Author(s):  
John Sinkovic ◽  
Mark Kempton
Keyword(s):  
Minimum Rank ◽  
Outerplanar Graphs

scholarly journals Relaxed game chromatic number of trees and outerplanar graphs

2004 ◽  
Vol 281 (1-3) ◽  
pp. 209-219 ◽  
Author(s):  
Wenjie He ◽  
Jiaojiao Wu ◽  
Xuding Zhu
Keyword(s):  
Chromatic Number ◽  
Outerplanar Graphs ◽  
Game Chromatic Number
Sign in / Sign up

Export Citation Format

Share Document