scholarly journals Independent sets in (P₆, diamond)-free graphs

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Raffaele Mosca
Keyword(s):  
Polynomial Time ◽  
Dominating Set ◽  
Minimum Weight ◽  
Independent Set ◽  
Independent Sets ◽  
Maximum Weight ◽  
Independent Set Problem ◽  
International Audience ◽  
Dominating Set Problem ◽  
Independent Dominating Set

Graphs and Algorithms International audience We prove that on the class of (P6,diamond)-free graphs the Maximum-Weight Independent Set problem and the Minimum-Weight Independent Dominating Set problem can be solved in polynomial time.

scholarly journals A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set

2012 ◽  
Vol Vol. 14 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Serge Gaspers ◽  
Mathieu Liedloff
Keyword(s):  
Dominating Set ◽  
Independent Set ◽  
Time Algorithm ◽  
Worst Case ◽  
Running Time ◽  
Running Time Analysis ◽  
Np Hard Problem ◽  
International Audience ◽  
Maximal Independent Sets ◽  
Independent Dominating Set

Graphs and Algorithms International audience An independent dominating set D of a graph G = (V,E) is a subset of vertices such that every vertex in V \ D has at least one neighbor in D and D is an independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum independent dominating set in a graph is an NP-hard problem. Whereas it is hard to cope with this problem using parameterized and approximation algorithms, there is a simple exact O(1.4423^n)-time algorithm solving the problem by enumerating all maximal independent sets. In this paper we improve the latter result, providing the first non trivial algorithm computing a minimum independent dominating set of a graph in time O(1.3569^n). Furthermore, we give a lower bound of \Omega(1.3247^n) on the worst-case running time of this algorithm, showing that the running time analysis is almost tight.

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.

scholarly journals Independent dominating set problem revisited

2015 ◽  
Vol 562 ◽  
pp. 1-22 ◽  
Author(s):  
Ching-Hao Liu ◽  
Sheung-Hung Poon ◽  
Jin-Yong Lin
Keyword(s):  
Dominating Set ◽  
Dominating Set Problem ◽  
Independent Dominating Set

scholarly journals New Instances for Maximum Weight Independent Set From a Vehicle Routing Application

2021 ◽  
Vol 2 (4) ◽  
Author(s):  
Yuanyuan Dong ◽  
Andrew V. Goldberg ◽  
Alexander Noe ◽  
Nikos Parotsidis ◽  
Mauricio G. C. Resende ◽  
...  
Keyword(s):  
Vehicle Routing ◽  
Real World ◽  
Vehicle Routing Problem ◽  
Independent Set ◽  
Maximum Weight ◽  
Routing Problem ◽  
Independent Set Problem ◽  
Maximum Weight Independent Set ◽  
Large Size

AbstractWe present a set of new instances of the maximum weight independent set problem. These instances are derived from a real-world vehicle routing problem and are challenging to solve in part because of their large size. We present instances with up to 881 thousand nodes and 383 million edges.

Independent domination number in Cayley digraphs of rectangular groups

2018 ◽  
Vol 10 (02) ◽  
pp. 1850024
Author(s):  
Nuttawoot Nupo ◽  
Sayan Panma
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Left Zero ◽  
Zero Semigroup ◽  
Left Zero Semigroup ◽  
Group A ◽  
Independent Domination ◽  
Independent Dominating Set ◽  
Rectangular Group

Let [Formula: see text] denote the Cayley digraph of the rectangular group [Formula: see text] with respect to the connection set [Formula: see text] in which the rectangular group [Formula: see text] is isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. An independent dominating set of [Formula: see text] is the independent set of elements in [Formula: see text] that can dominate the whole elements. In this paper, we investigate the independent domination number of [Formula: see text] and give more results on Cayley digraphs of left groups and right groups which are specific cases of rectangular groups. Moreover, some results of the path independent domination number of [Formula: see text] are also shown.

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