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 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 On the Total Outer k-Independent Domination Number of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 194 ◽  
Author(s):  
Abel Cabrera-Martínez ◽  
Juan Carlos Hernández-Gómez ◽  
Ernesto Parra-Inza ◽  
José María Sigarreta Almira
Keyword(s):  
Maximum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Np Hard Problem ◽  
Independent Domination ◽  
Independent Dominating Set ◽  
Computational Properties

A set of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex in such a set. We say that a total dominating set D is a total outer k-independent dominating set of G if the maximum degree of the subgraph induced by the vertices that are not in D is less or equal to k − 1 . The minimum cardinality among all total outer k-independent dominating sets is the total outer k-independent domination number of G. In this article, we introduce this parameter and begin with the study of its combinatorial and computational properties. For instance, we give several closed relationships between this novel parameter and other ones related to domination and independence in graphs. In addition, we give several Nordhaus–Gaddum type results. Finally, we prove that computing the total outer k-independent domination number of a graph G is an NP-hard problem.

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.

scholarly journals Partial Quicksort and Quickpartitionsort

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Conrado Martínez ◽  
Uwe Rösler
Keyword(s):  
Lower Bound ◽  
Time Analysis ◽  
Information Theoretic ◽  
Running Time ◽  
Running Time Analysis ◽  
International Audience

International audience Partial Quicksort sorts the $l$ smallest elements in a list of length $n$. We provide a complete running time analysis for this combination of Find and Quicksort. Further we give some optimal adapted versions, called Partition Quicksort, with an asymptotic running time $c_1l\ln l+c_2l+n+o(n)$. The constant $c_1$ can be as small as the information theoretic lower bound $\log_2 e$.

scholarly journals Common Independence in Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1411
Author(s):  
Magda Dettlaff ◽  
Magdalena Lemańska ◽  
Jerzy Topp
Keyword(s):  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Independent Subset ◽  
Block Graphs ◽  
Independent Domination ◽  
The Common ◽  
Independent Dominating Set

The cardinality of a largest independent set of G, denoted by α(G), is called the independence number of G. The independent domination number i(G) of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by αc(G), as the greatest integer r such that every vertex of G belongs to some independent subset X of VG with |X|≥r. The common independence number αc(G) of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality αc(G) in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities i(G)≤αc(G)≤α(G). In this paper, we characterize the trees T for which i(T)=αc(T), and the block graphs G for which αc(G)=α(G).

scholarly journals Fast separation in a graph with an excluded minor

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Bruce Reed ◽  
David R. Wood
Keyword(s):  
Open Problem ◽  
Time Algorithm ◽  
Total Weight ◽  
Running Time ◽  
Fast Separation ◽  
Edge Graph ◽  
Excluded Minor ◽  
International Audience ◽  
Vertex Sets

International audience Let $G$ be an $n$-vertex $m$-edge graph with weighted vertices. A pair of vertex sets $A,B \subseteq V(G)$ is a $\frac{2}{3} - \textit{separation}$ of $\textit{order}$ $|A \cap B|$ if $A \cup B = V(G)$, there is no edge between $A \backslash B$ and $B \backslash A$, and both $A \backslash B$ and $B \backslash A$ have weight at most $\frac{2}{3}$ the total weight of $G$. Let $\ell \in \mathbb{Z}^+$ be fixed. Alon, Seymour and Thomas [$\textit{J. Amer. Math. Soc.}$ 1990] presented an algorithm that in $\mathcal{O}(n^{1/2}m)$ time, either outputs a $K_\ell$-minor of $G$, or a separation of $G$ of order $\mathcal{O}(n^{1/2})$. Whether there is a $\mathcal{O}(n+m)$ time algorithm for this theorem was left as open problem. In this paper, we obtain a $\mathcal{O}(n+m)$ time algorithm at the expense of $\mathcal{O}(n^{2/3})$ separator. Moreover, our algorithm exhibits a tradeoff between running time and the order of the separator. In particular, for any given $\epsilon \in [0,\frac{1}{2}]$, our algorithm either outputs a $K_\ell$-minor of $G$, or a separation of $G$ with order $\mathcal{O}(n^{(2-\epsilon )/3})$ in $\mathcal{O}(n^{1+\epsilon} +m)$ time.

scholarly journals Maximal independent sets in bipartite graphs with at least one cycle

2013 ◽  
Vol Vol. 15 no. 2 (Graph Theory) ◽  
Author(s):  
Shuchao Li ◽  
Huihui Zhang ◽  
Xiaoyan Zhang
Keyword(s):  
Graph Theory ◽  
Independent Set ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Proper Subset ◽  
Maximal Independent Set ◽  
Extremal Graphs ◽  
International Audience ◽  
Maximal Independent Sets ◽  
Disconnected Graphs

Graph Theory International audience A maximal independent set is an independent set that is not a proper subset of any other independent set. Liu [J.Q. Liu, Maximal independent sets of bipartite graphs, J. Graph Theory, 17 (4) (1993) 495-507] determined the largest number of maximal independent sets among all n-vertex bipartite graphs. The corresponding extremal graphs are forests. It is natural and interesting for us to consider this problem on bipartite graphs with cycles. Let \mathscrBₙ (resp. \mathscrBₙ') be the set of all n-vertex bipartite graphs with at least one cycle for even (resp. odd) n. In this paper, the largest number of maximal independent sets of graphs in \mathscrBₙ (resp. \mathscrBₙ') is considered. Among \mathscrBₙ the disconnected graphs with the first-, second-, \ldots, \fracn-22-th largest number of maximal independent sets are characterized, while the connected graphs in \mathscrBₙ having the largest, the second largest number of maximal independent sets are determined. Among \mathscrBₙ' graphs have the largest number of maximal independent sets are identified.

scholarly journals Independent Domination Subdivision in Graphs

2021 ◽  
Author(s):  
Ammar Babikir ◽  
Magda Dettlaff ◽  
Michael A. Henning ◽  
Magdalena Lemańska
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Independent Domination ◽  
Delta G ◽  
Independent Dominating Set ◽  
Domination Subdivision Number

AbstractA set S of vertices in a graph G is a dominating set if every vertex not in S is ad jacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. The independent domination subdivision number $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the independent domination number. We show that for every connected graph G on at least three vertices, the parameter $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is well defined and differs significantly from the well-studied domination subdivision number $$\mathrm{sd_\gamma }(G)$$ sd γ ( G ) . For example, if G is a block graph, then $$\mathrm{sd_\gamma }(G) \le 3$$ sd γ ( G ) ≤ 3 , while $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) can be arbitrary large. Further we show that there exist connected graph G with arbitrarily large maximum degree $$\Delta (G)$$ Δ ( G ) such that $$ \hbox {sd}_{\mathrm{i}}(G) \ge 3 \Delta (G) - 2$$ sd i ( G ) ≥ 3 Δ ( G ) - 2 , in contrast to the known result that $$\mathrm{sd_\gamma }(G) \le 2 \Delta (G) - 1$$ sd γ ( G ) ≤ 2 Δ ( G ) - 1 always holds. Among other results, we present a simple characterization of trees T with $$ \hbox {sd}_{\mathrm{i}}(T) = 1$$ sd i ( T ) = 1 .

scholarly journals Enumeration and Random Generation of Concurrent Computations

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Olivier Bodini ◽  
Antoine Genitrini ◽  
Frédéric Peschanski
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Random Generation ◽  
Worst Case ◽  
Analytic Combinatorics ◽  
Mean Width ◽  
Concurrent Processes ◽  
International Audience ◽  
The Mean

International audience In this paper, we study the shuffle operator on concurrent processes (represented as trees) using analytic combinatorics tools. As a first result, we show that the mean width of shuffle trees is exponentially smaller than the worst case upper-bound. We also study the expected size (in total number of nodes) of shuffle trees. We notice, rather unexpectedly, that only a small ratio of all nodes do not belong to the last two levels. We also provide a precise characterization of what ``exponential growth'' means in the case of the shuffle on trees. Two practical outcomes of our quantitative study are presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random generation of concurrent runs.

An efficient incremental DFA minimization algorithm

2003 ◽  
Vol 9 (1) ◽  
pp. 49-64 ◽  
Author(s):  
BRUCE W. WATSON ◽  
JAN DACIUK
Keyword(s):  
Large Class ◽  
Finite Automata ◽  
Time Algorithm ◽  
The Other ◽  
Minimization Algorithm ◽  
Worst Case ◽  
Deterministic Finite Automata ◽  
Running Time ◽  
The Third ◽  
Minimization Algorithms

In this paper, we present a new Deterministic Finite Automata (DFA) minimization algorithm. The algorithm is incremental – it may be halted at any time, yielding a partially-minimized automaton. All of the other (known) minimization algorithms have intermediate results which are not useable for partial minimization. Since the first algorithm is easily understood but inefficient, we consider three practical and effective optimizations. The first two optimizations do not affect the asymptotic worst-case running time – though they perform well on a large class of automata. The third optimization yields an quadratic-time algorithm which is competitive with the previously known ones.

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