SIAM Journal on Discrete Mathematics

10.1137/sidma ◽  
2014 ◽  
Keyword(s):  
Discrete Mathematics ◽  
Siam Journal

scholarly journals Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

2019 ◽  
Vol 30 (02) ◽  
pp. 197-230 ◽  
Author(s):  
Markus Chimani ◽  
Giuseppe Di Battista ◽  
Fabrizio Frati ◽  
Karsten Klein
Keyword(s):  
Polynomial Time ◽  
Spanning Trees ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Negative Instance ◽  
Discrete Mathematics ◽  
Connected Components ◽  
Linear Time Algorithm ◽  
Siam Journal

In this paper, we show a polynomial-time algorithm for testing [Formula: see text]-planarity of embedded flat clustered graphs with at most two vertices per cluster on each face. Our result is based on a reduction to the planar set of spanning trees in topological multigraphs (pssttm) problem, which is defined as follows. Given a (non-planar) topological multigraph [Formula: see text] with [Formula: see text] connected components [Formula: see text], do spanning trees of [Formula: see text] exist such that no two edges in any two spanning trees cross? Kratochvíl et al. [SIAM Journal on Discrete Mathematics, 4(2): 223–244, 1991] proved that the problem is NP-hard even if [Formula: see text]; on the other hand, Di Battista and Frati presented a linear-time algorithm to solve the pssttm problem for the case in which [Formula: see text] is a [Formula: see text]-planar topological multigraph [Journal of Graph Algorithms and Applications, 13(3): 349–378, 2009]. For any embedded flat clustered graph [Formula: see text], an instance [Formula: see text] of the pssttm problem can be constructed in polynomial time such that [Formula: see text] is [Formula: see text]-planar if and only if [Formula: see text] admits a solution. We show that, if [Formula: see text] has at most two vertices per cluster on each face, then it can be tested in polynomial time whether the corresponding instance [Formula: see text] of the pssttm problem is positive or negative. Our strategy for solving the pssttm problem on [Formula: see text] is to repeatedly perform a sequence of tests, which might let us conclude that [Formula: see text] is a negative instance, and simplifications, which might let us simplify [Formula: see text] by removing or contracting some edges. Most of these tests and simplifications are performed “locally”, by looking at the crossings involving a single edge or face of a connected component [Formula: see text] of [Formula: see text]; however, some tests and simplifications have to consider certain global structures in [Formula: see text], which we call [Formula: see text]-donuts. If no test concludes that [Formula: see text] is a negative instance of the pssttm problem, then the simplifications eventually transform [Formula: see text] into an equivalent [Formula: see text]-planar topological multigraph on which we can apply the cited linear-time algorithm by Di Battista and Frati.

scholarly journals A note on PM-compact $ K_4 $-free bricks

2021 ◽  
Vol 7 (3) ◽  
pp. 3648-3652
Author(s):  
Jinqiu Zhou ◽  
◽  
Qunfang Li ◽  
Keyword(s):  
Connected Graph ◽  
Perfect Matching ◽  
Building Blocks ◽  
Discrete Mathematics ◽  
Siam Journal ◽  
Multiple Edges ◽  
Generation Procedure

<abstract><p>A 3-connected graph is a <italic>brick</italic> if the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching covered graphs. Lovász (Combinatorica, 3 (1983), 105-117) showed that every brick is $ K_4 $-based or $ \overline{C}_6 $-based. A brick is <italic>$ K_4 $-free</italic> (respectively, <italic>$ \overline{C}_6 $-free</italic>) if it is not $ K_4 $-based (respectively, $ \overline{C}_6 $-based). Recently, Carvalho, Lucchesi and Murty (SIAM Journal on Discrete Mathematics, 34(3) (2020), 1769-1790) characterised the PM-compact $ \overline{C}_6 $-free bricks. In this note, we show that, by using the brick generation procedure established by Norine and Thomas (J Combin Theory Ser B, 97 (2007), 769-817), the only PM-compact $ K_4 $-free brick is $ \overline{C}_6 $, up to multiple edges.</p></abstract>

Learning Discrete Mathematics with ISETL

1989 ◽  
Author(s):  
Nancy Baxter ◽  
Ed Dubinsky ◽  
Gary Levin
Keyword(s):  
Discrete Mathematics

scholarly journals Quadruple Roman Domination in Trees

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1318
Author(s):  
Zheng Kou ◽  
Saeed Kosari ◽  
Guoliang Hao ◽  
Jafar Amjadi ◽  
Nesa Khalili
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Theoretical Computer Science ◽  
Upper And Lower Bounds ◽  
Discrete Mathematics ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function

This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph G is a function from the vertex set V of G to the set {0,1,2,…,k+1} if for any vertex u∈V with f(u)<k, ∑x∈N(u)∪{u}f(x)≥|{x∈N(u):f(x)≥1}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value Σv∈Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number γ[kR](G) of G. In this paper, we establish sharp upper and lower bounds on γ[4R](T) for nontrivial trees T and characterize extremal trees.

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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