A Linear-Time Algorithm for Constructing an Optimal Node-Search Strategy of a Tree

In this paper, we consider the problem of finding the minimum number of searchers to sweep networks/graphs with special topological structures. Such a number is called the search number. We first study graphs, which contain only one cycle, and present a linear time algorithm to compute the vertex separation and the optimal layout of such graphs; by a linear-time transformation, we can find the search number of this kind of graphs in linear time. We also investigate graphs, in which every vertex lies on at most one cycle and each cycle contains at most three vertices of degree more than two, and we propose a linear time algorithm to compute their search number and optimal search strategy. We prove explicit formulas for the search number of the graphs obtained from complete k-ary trees by replacing vertices by cycles. We also present some results on approximation algorithms.

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Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽

10.3390/math9030293 ◽

2021 ◽

Vol 9 (3) ◽

pp. 293

Author(s):

Xinyue Liu ◽

Huiqin Jiang ◽

Pu Wu ◽

Zehui Shao

Keyword(s):

Linear Time ◽

Minimum Weight ◽

Domination Number ◽

Time Algorithm ◽

Simple Graph ◽

Linear Time Algorithm ◽

Domination Problem ◽

Np Complete ◽

Chordal Bipartite Graphs ◽

Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

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Three-coloring triangle-free graphs on surfaces VII. A linear-time algorithm

Journal of Combinatorial Theory Series B ◽

10.1016/j.jctb.2021.07.002 ◽

2021 ◽

Author(s):

Zdeněk Dvořák ◽

Daniel Král' ◽

Robin Thomas

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Graphs On Surfaces

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Nearly Linear Time Algorithm for Mean Hitting Times of Random Walks on a Graph

Proceedings of the 13th International Conference on Web Search and Data Mining ◽

10.1145/3336191.3371777 ◽

2020 ◽

Author(s):

Zuobai Zhang ◽

Wanyue Xu ◽

Zhongzhi Zhang

Keyword(s):

Random Walks ◽

Linear Time ◽

Time Algorithm ◽

Hitting Times ◽

Linear Time Algorithm

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A Linear time algorithm for deciding security

10.1109/sfcs.1976.1 ◽

1976 ◽

Cited By ~ 33

Author(s):

A. K. Jones ◽

R. J. Lipton ◽

L. Snyder

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Linear Time Algorithm

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AN IMPROVED ALGORITHM FOR FINDING TREE DECOMPOSITIONS OF SMALL WIDTH

International Journal of Foundations of Computer Science ◽

10.1142/s0129054100000247 ◽

2000 ◽

Vol 11 (03) ◽

pp. 365-371 ◽

Cited By ~ 33

Author(s):

LJUBOMIR PERKOVIĆ ◽

BRUCE REED

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Tree Decomposition ◽

Linear Time Algorithm ◽

Input Graph ◽

Primary Motivation ◽

Small Width ◽

Tree Decompositions ◽

Improved Algorithm

We present a modification of Bodlaender's linear time algorithm that, for constant k, determine whether an input graph G has treewidth k and, if so, constructs a tree decomposition of G of width at most k. Our algorithm has the following additional feature: if G has treewidth greater than k then a subgraph G′ of G of treewidth greater than k is returned along with a tree decomposition of G′ of width at most 2k. A consequence is that the fundamental disjoint rooted paths problem can now be solved in O(n2) time. This is the primary motivation of this paper.

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