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On the maximum number of minimal connected dominating sets in convex bipartite graphs [article]

Mohamed Yosri Sayadi
2019 arXiv   pre-print
We present an algorithm to enumerate all minimal connected dominating sets of convex bipartite graphs in time O(1.7254^n) where n is the number of vertices of the input graph.  ...  Hence, to the best of our knowledge, there is no known input-sensitive algorithm for enumerating minimal dominating sets, or one of their related sets, in bipartite graphs better than that of general graphs  ...  However, the minimum connected 1:2 On the maximum number of minimal connected dominating sets in convex bipartite graphs dominating set problem is tractable for convex bipartite graphs and it can be solved  ... 
arXiv:1908.02174v1 fatcat:q353hmpvovg5jeiqjbn7cdsdiu

Page 2761 of Mathematical Reviews Vol. , Issue 98E [page]

1998 Mathematical Reviews  
Mynhardt (SA-PRTR; Pretoria) 98e:05070 05C35 Yu, Bo |Yu, Bo*| (3-WTRL-B; Waterloo, ON) Convexity of minimal total dominating functions in graphs. (English summary) J.  ...  This paper studies the convexity of minimal total dominating functions (MTDF). An MTDF / is called universal if convex combinations of f and any other MTDF are minimal.  ... 

Red Domination in Perfect Elimination Bipartite Graphs [article]

Nesrine Abbas
2022 arXiv   pre-print
We present a tight upper bound on the number of such sets in bipartite graphs, and show that we can calculate that number in linear time for convex bipartite graphs.  ...  The k red domination problem for a bipartite graph G=(X,Y,E) is to find a subset D ⊆ X of cardinality at most k that dominates vertices of Y.  ...  In the section that follows we present a tight upper bound on the number of vertices in a red dominating set in a bipartite graph, show how to calculate the number of such sets in linear time for convex  ... 
arXiv:2203.10630v1 fatcat:aawxkajsdvh2ndngynrswyez7q

Page 7219 of Mathematical Reviews Vol. , Issue 96m [page]

1996 Mathematical Reviews  
Summary: “An edge dominating set E of a graph G is a subset of the edge set of G such that every edge not in E is adjacent to at least one edge in E.  ...  The edge domination number of G is the cardinality of a minimum edge dominating set of G.  ... 

Page 2849 of Mathematical Reviews Vol. , Issue 86g [page]

1986 Mathematical Reviews  
For a graph G, a subset of vertices D is a dominating set if every vertex not in D is adjacent to some vertex in D. The domination number, denoted 4(G), is the order of the smallest dominating set.  ...  The number of d-convex sets in a graph. (Russian) Izv. Akad. Nauk Moldav. SSR Ser. Fiz.-Tekhn. Mat.  ... 

Page 6632 of Mathematical Reviews Vol. , Issue 92m [page]

1992 Mathematical Reviews  
The k-independence number a;(G) of G is the maximum number of vertices of a k-independent set in G.  ...  An extremal graph F has the maximum number of edges in a graph feasible for (G, x); its complement F is minimal for (G, x).  ... 

Page 1301 of Mathematical Reviews Vol. , Issue 84d [page]

1984 Mathematical Reviews  
On the minimum dominating pair number of a class of graphs. Caribbean J. Math. 1 (1982), no. 2, 73-76.  ...  In particular, G is the unique graph which has the maximum number of spanning trees among all the graphs with wv vertices and e edges.  ... 

On the complexity of the dominating induced matching problem in hereditary classes of graphs

Domingos M. Cardoso, Nicholas Korpelainen, Vadim V. Lozin
2011 Discrete Applied Mathematics  
In the present paper, we identify a critical class for this problem (i.e., a class lying on a "boundary" separating difficult instances of the problem from polynomially solvable ones) and derive a number  ...  The dominating induced matching problem, also known as efficient edge domination, is the problem of determining whether a graph has an induced matching that dominates every edge of the graph.  ...  Cardoso was supported by the Centre of Research on Optimization and Control from the Fundação para a Ciência e Tecnologia cofinanced by the European Community Fund FEDER/POCI/2010 and the research of Vadim  ... 
doi:10.1016/j.dam.2010.03.011 fatcat:q236falzp5e3rmqa3bxaolpddy

Introduction

MartinCharles Golumbic, Renu Laskar
1993 Discrete Applied Mathematics  
It is known that for chordal graphs, the size of a largest minimal dominating set, the size of a maximum independent set, and the size of a maximum irredundant set are all equal.  ...  In their paper, Elmallah and Stewart provide polynomial time algorithms for the minimum dominating set problem and the maximum r-independent set problem on the family of intersection graphs of chords of  ... 
doi:10.1016/0166-218x(93)90217-c fatcat:iklvodekpnho3nyhvmz6acnvuy

Enumeration of Enumeration Algorithms [article]

Kunihiro Wasa
2016 arXiv   pre-print
In this paper, we enumerate enumeration problems and algorithms. This survey is under construction. If you know some results not in this survey or there is anything wrong, please let me know.  ...  Reference [115] 2.12.15 Enumeration of all minimal dominating sets in a chordal bipartite graph Input A chordal bipartite graph G. Output All minimal dominating sets in G.  ...  Reference [95] 2.12.10 Enumeration of all minimal edge dominating sets in a bipartite graph Input A bipartite graph G = (V, E). Output All minimal edge dominating sets in G.  ... 
arXiv:1605.05102v2 fatcat:bwmhesagkje5hmynqyij7jv224

Algorithmic aspects of Roman {3}-domination in graphs

Chakradhar Padamutham, Dr. P. Venkata Subba Reddy
2022 Reserche operationelle  
In this paper, first we show that the R3DP is NP-complete for chordal graphs, planar graphs and for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs  ...  vertex $u \in G$, where $N_G(u)$ is the set of vertices adjacent to $u$ in $G$, is called a \textit{ Roman \{3\}-dominating function} (R3DF) of $G$.  ...  The domination number is the minimum cardinality of a dominating set in G and is denoted by γ(G) [25] . The concept of Roman domination was introduced in 2004 by Cockayne et al. in [1] .  ... 
doi:10.1051/ro/2022106 fatcat:3t3to5uakva7jdo7ou3lip3wha

Page 7560 of Mathematical Reviews Vol. , Issue 98M [page]

1998 Mathematical Reviews  
The authors are interested in determining the maximum possible domination number of a graph (not necessarily connected) with a given number of vertices, say n, and with a specified minimum degree.  ...  An asymptotic bound is derived for the number of connected graphs with any number of cutpoints.” 98m:05094 05C35 Alon, Noga (IL-TLAV; Tel Aviv); Halperin, Eran (I1L-TLAV-C; Tel Aviv) Bipartite subgraphs  ... 

Hamiltonicity in Convex Bipartite Graphs [article]

P. Kowsika, V. Divya, N. Sadagopan
2018 arXiv   pre-print
For a connected graph, the Hamiltonian cycle (path) is a simple cycle (path) that spans all the vertices in the graph.  ...  We present an infinite set of counterexamples in support of our claim.  ...  Acknowledgements: The authors wish to appreciate and thank N Narayanan (IIT Madras), C Venkata Praveena (IIITDM Kancheepuram) and P Renjith (IIIT Kottayam) for sharing their insights on this problem.  ... 
arXiv:1809.06113v1 fatcat:xsbfav6tpna3no622lq7wfomkm

Page 6169 of Mathematical Reviews Vol. , Issue 90K [page]

1990 Mathematical Reviews  
Three inequalities between the domination number (the maximum number of vertices of a dominating set) and the chromatic number of a finite undirected graph are presented.  ...  The note is concerned with the problem of finding a partition of a given set of vertices A into a minimum possible number of convex sets.  ... 

Restrictions of Minimum Spanner Problems

G. Venkatesan, U. Rotics, M.S. Madanlal, J.A. Makowsky, C.Pandu Rangan
1997 Information and Computation  
In this paper, we completely settle the complexity status of this problem for various values of t, on chordal graphs, split graphs, bipartite graphs and convex bipartite graphs.  ...  Finally, we provide approximation algorithms for the bandwidth minimization problem on convex bipartite graphs and split graphs using the notion of tree spanners. ] 1997 Academic Press article no.  ...  graphs, which also appears in [21] ; [22] was subsequently replaced by the current paper with five authors.  ... 
doi:10.1006/inco.1997.2641 fatcat:wdqj773b4bbs7mmg24yfzgoxwq
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