Semitotal Domination on AT-Free Graphs and Circle Graphs

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

Symmetry ◽

10.3390/sym13061036 ◽

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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New clique and independent set algorithms for circle graphs

Discrete Applied Mathematics ◽

10.1016/0166-218x(92)90200-t ◽

1992 ◽

Vol 36 (1) ◽

pp. 1-24 ◽

Cited By ~ 21

Author(s):

Alberto Apostolico ◽

Mikhail J. Atallah ◽

Susanne E. Hambrusch

Keyword(s):

Independent Set ◽

Circle Graphs

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MINIMUM WEIGHT FEEDBACK VERTEX SETS IN CIRCLE n-GON GRAPHS AND CIRCLE TRAPEZOID GRAPHS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830911001243 ◽

2011 ◽

Vol 03 (03) ◽

pp. 323-336 ◽

Cited By ~ 2

Author(s):

FANICA GAVRIL

Keyword(s):

Minimum Weight ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Circle Graphs ◽

Intersection Model ◽

The Family ◽

Positive Weights ◽

Vertex Set ◽

Vertex Sets ◽

Trapezoid Graphs

A circle n-gon is the region between n or fewer non-crossing chords of a circle, no chord connecting the arcs between two other chords; the sides of a circle n-gon are either chords or arcs of the circle. A circle n-gon graph is the intersection graph of a family of circle n-gons in a circle. The family of circle trapezoid graphs is exactly the family of circle 2-gon graphs and the family of circle graphs is exactly the family of circle 1-gon graphs. The family of circle n-gon graphs contains the polygon-circle graphs which have an intersection representation by circle polygons, each polygon with at most n chords. We describe a polynomial time algorithm to find a minimum weight feedback vertex set, or equivalently, a maximum weight induced forest, in a circle n-gon graph with positive weights, when its intersection model by n-gon-interval-filaments is given.

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