ALMOST OVERLAP-FREE WORDS AND THE WORD PROBLEM FOR THE FREE BURNSIDE SEMIGROUP SATISFYING x2 = x3

We study the word problem for the free Burnside semigroup satisfying x2 = x3 and having two generators. The elements of this semigroup are classes of equivalent words. A natural way to solve the word problem is to select a unique "canonical" representative for each equivalence class. We prove that overlap-free words and "almost" overlap-free words can serve as canonical representatives of their equivalence classes. We show that such a word in a given class, if any, can be efficiently found. As a result, we construct a linear-time algorithm that partially solves the word problem for the semigroup under consideration.

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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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