scholarly journals Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes

2021 ◽  
Vol 35 (1) ◽  
pp. 105-151
Author(s):  
Archontia Giannopoulou ◽  
Michał Pilipczuk ◽  
Jean-Florent Raymond ◽  
Dimitrios M. Thilikos ◽  
Marcin Wrochna
Keyword(s):  
Closed Graph ◽  
Edge Deletion ◽  
Graph Classes

scholarly journals Minor-Closed Graph Classes with Bounded Layered Pathwidth

2020 ◽  
Vol 34 (3) ◽  
pp. 1693-1709
Author(s):  
Vida Dujmović ◽  
David Eppstein ◽  
Gwenaël Joret ◽  
Pat Morin ◽  
David R. Wood
Keyword(s):  
Closed Graph ◽  
Graph Classes

scholarly journals Layered separators in minor-closed graph classes with applications

2017 ◽  
Vol 127 ◽  
pp. 111-147 ◽  
Author(s):  
Vida Dujmović ◽  
Pat Morin ◽  
David R. Wood
Keyword(s):  
Closed Graph ◽  
Graph Classes

scholarly journals Reconfiguring Dominating Sets in Minor-Closed Graph Classes

2021 ◽  
Author(s):  
Dieter Rautenbach ◽  
Johannes Redl
Keyword(s):  
Positive Constant ◽  
Negative Integer ◽  
Dominating Sets ◽  
Closed Graph ◽  
Hereditary Class ◽  
Graph Classes ◽  
Large Order ◽  
Toroidal Graphs

AbstractFor a graph G, two dominating sets D and $$D'$$ D ′ in G, and a non-negative integer k, the set D is said to k-transform to $$D'$$ D ′ if there is a sequence $$D_0,\ldots ,D_\ell $$ D 0 , … , D ℓ of dominating sets in G such that $$D=D_0$$ D = D 0 , $$D'=D_\ell $$ D ′ = D ℓ , $$|D_i|\le k$$ | D i | ≤ k for every $$i\in \{ 0,1,\ldots ,\ell \}$$ i ∈ { 0 , 1 , … , ℓ } , and $$D_i$$ D i arises from $$D_{i-1}$$ D i - 1 by adding or removing one vertex for every $$i\in \{ 1,\ldots ,\ell \}$$ i ∈ { 1 , … , ℓ } . We prove that there is some positive constant c and there are toroidal graphs G of arbitrarily large order n, and two minimum dominating sets D and $$D'$$ D ′ in G such that Dk-transforms to $$D'$$ D ′ only if $$k\ge \max \{ |D|,|D'|\}+c\sqrt{n}$$ k ≥ max { | D | , | D ′ | } + c n . Conversely, for every hereditary class $$\mathcal{G}$$ G that has balanced separators of order $$n\mapsto n^\alpha $$ n ↦ n α for some $$\alpha <1$$ α < 1 , we prove that there is some positive constant C such that, if G is a graph in $$\mathcal{G}$$ G of order n, and D and $$D'$$ D ′ are two dominating sets in G, then Dk-transforms to $$D'$$ D ′ for $$k=\max \{ |D|,|D'|\}+\lfloor Cn^\alpha \rfloor $$ k = max { | D | , | D ′ | } + ⌊ C n α ⌋ .

scholarly journals Reconfiguration on Nowhere Dense Graph Classes

10.37236/7458 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Sebastian Siebertz
Keyword(s):  
Minimum Distance ◽  
Dominating Set ◽  
Independent Set ◽  
Closed Graph ◽  
Independent Set Problem ◽  
Graph Classes ◽  
Dominating Set Problem ◽  
Feasible Solutions ◽  
Dense Class ◽  
Kernelization Algorithms

Let $\mathcal{Q}$ be a vertex subset problem on graphs. In a reconfiguration variant of $\mathcal{Q}$ we are given a graph $G$ and two feasible solutions $S_s, S_t\subseteq V(G)$ of $\mathcal{Q}$ with $|S_s|=|S_t|=k$. The problem is to determine whether there exists a sequence $S_1,\ldots,S_n$ of feasible solutions, where $S_1=S_s$, $S_n=S_t$, $|S_i|\leq k\pm 1$, and each $S_{i+1}$ results from $S_i$, $1\leq i<n$, by the addition or removal of a single vertex.We prove that for every nowhere dense class of graphs and for every integer $r\geq 1$ there exists a polynomial $p_r$ such that the reconfiguration variants of the distance-$r$ independent set problem and the distance-$r$ dominating set problem admit kernels of size $p_r(k)$. If $k$ is equal to the size of a minimum distance-$r$ dominating set, then for any fixed $\epsilon>0$ we even obtain a kernel of almost linear size $\mathcal{O}(k^{1+\epsilon})$. We then prove that if a class $\mathcal{C}$ is somewhere dense and closed under taking subgraphs, then for some value of $r\geq 1$ the reconfiguration variants of the above problems on $\mathcal{C}$ are $\mathsf{W}[1]$-hard (and in particular we cannot expect the existence of kernelization algorithms). Hence our results show that the limit of tractability for the reconfiguration variants of the distance-$r$ independent set problem and distance-$r$ dominating set problem on subgraph closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.

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