scholarly journals Complexity of Counting Subgraphs: Only the Boundedness of the Vertex-Cover Number Counts

2014 ◽  
Author(s):  
Radu Curticapean ◽  
Daniel Marx
Keyword(s):  
Vertex Cover ◽  
Number Counts

Complexity of the Maximum k-Path Vertex Cover Problem

2020 ◽  
Vol E103.A (10) ◽  
pp. 1193-1201
Author(s):  
Eiji MIYANO ◽  
Toshiki SAITOH ◽  
Ryuhei UEHARA ◽  
Tsuyoshi YAGITA ◽  
Tom C. van der ZANDEN
Keyword(s):  
Vertex Cover ◽  
Vertex Cover Problem ◽  
Cover Problem

scholarly journals Approximation algorithm for minimum connected 3-path vertex cover

2020 ◽  
Vol 287 ◽  
pp. 77-84
Author(s):  
Pengcheng Liu ◽  
Zhao Zhang ◽  
Xianyue Li ◽  
Weili Wu
Keyword(s):  
Approximation Algorithm ◽  
Vertex Cover

scholarly journals From the Quasi-Total Strong Differential to Quasi-Total Italian Domination in Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1036
Author(s):  
Abel Cabrera Martínez ◽  
Alejandro Estrada-Moreno ◽  
Juan Alberto Rodríguez-Velázquez
Keyword(s):  
Vertex Cover ◽  
Domination Number ◽  
Theoretical Computer Science ◽  
Discrete Mathematics ◽  
Special Issue ◽  
Total Domination Number ◽  
Theoretical Computer ◽  
Graph Parameters ◽  
Semitotal Domination ◽  
Domination In Graphs

This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. Given a vertex x∈V(G) of a graph G, the neighbourhood of x is denoted by N(x). The neighbourhood of a set X⊆V(G) is defined to be N(X)=⋃x∈XN(x), while the external neighbourhood of X is defined to be Ne(X)=N(X)∖X. Now, for every set X⊆V(G) and every vertex x∈X, the external private neighbourhood of x with respect to X is defined as the set Pe(x,X)={y∈V(G)∖X:N(y)∩X={x}}. Let Xw={x∈X:Pe(x,X)≠⌀}. The strong differential of X is defined to be ∂s(X)=|Ne(X)|−|Xw|, while the quasi-total strong differential of G is defined to be ∂s*(G)=max{∂s(X):X⊆V(G)andXw⊆N(X)}. We show that the quasi-total strong differential is closely related to several graph parameters, including the domination number, the total domination number, the 2-domination number, the vertex cover number, the semitotal domination number, the strong differential, and the quasi-total Italian domination number. As a consequence of the study, we show that the problem of finding the quasi-total strong differential of a graph is NP-hard.

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