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### Factor domination and minimum degree

P. Dankelmann, R.C. Laskar
2003 Discrete Mathematics
In this paper, we give bounds on (F1; F2; : : : ; F k ) in terms of the minimum degrees of the Fi.  ...  The cardinality of a smallest such set is called the factor domination number of F1; F2; : : : ; F k and denoted by (F1; F2; : : : ; F k ).  ...  In this paper, we give bounds on the factor domination number in terms of the order and the minimum degrees of the factors. Our notation is as follows.  ...

### Exact Algorithms for Finding the Minimum Independent Dominating Set in Graphs [chapter]

Chunmei Liu, Yinglei Song
2006 Lecture Notes in Computer Science
In this paper, we consider the Minimum Independent Dominating Set problem and develop exact exponential algorithms that break the trivial developed to solve this problem on general graphs.  ...  For sparse graphs, e.g. graphs with degree bounded by 3 and 4, we show that a few new branching techniques can be applied to these graphs and the resulting algorithms have time complexities O * (2 0.465  ...  Acknowledgment We thank the anonymous reviewers for their comments and suggestions on an earlier version of the paper.  ...

### Simultaneous graph parameters: Factor domination and factor total domination

Peter Dankelmann, Michael A. Henning, Wayne Goddard, Renu Laskar
2006 Discrete Mathematics
A subset S ⊆ V is a factor dominating set if in every F i every vertex not in S is adjacent to a vertex in S, and a factor total dominating set if in every F i every vertex in V is adjacent to a vertex  ...  The cardinality of a smallest such set is the factor (total) domination number. In this note we investigate bounds on the factor (total) domination number.  ...  Surprisingly perhas, the best bounds for minimum degree 3 and minimum degree 4 come from two different sources.  ...

### Star-Uniform Graphs

Mikio Kano, Yunjian Wu, Qinglin Yu
2010 Graphs and Combinatorics
In this paper, we show that a graph is star-uniform if and only if has equal domination and matching number.  ...  Motivated by the minimum cost spanning tree and the optimal assignment problems, Hartnell and Rall posed an open problem to characterize all the star-uniform graphs.  ...  Acknowledgments The authors are indebted to the anonymous referees for their many helpful and constructive suggestions.  ...

### Complexity of majority monopoly and signed domination problems

Sounaka Mishra
2012 Journal of Discrete Algorithms
We show that minimum majority monopoly problem is APX-complete for graphs with degree at most 3 and at least 2 and minimum signed domination problem is APX-complete, for 3-regular graphs.  ...  We consider approximability of two natural variants of classical dominating set problem, namely minimum majority monopoly and minimum signed domination.  ...  Acknowledgement The author thanks the anonymous referees for their valuable comments and suggestions that helped to improve the clarity, correctness and presentation of the proofs.  ...

### Page 2455 of Mathematical Reviews Vol. , Issue 2000d [page]

2000 Mathematical Reviews
Let k >2 and let G be a graph of order m with minimum degree at least k, and with n > 8k — 16 for even n, and n > 6k — 13 for odd n.  ...  Those vertices that belong to no minimum dominating set are also investigated and characterized when the graphs are trees.  ...

### Fast Algorithms for min independent dominating set [chapter]

Nicolas Bourgeois, Bruno Escoffier, Vangelis Th. Paschos
2010 Lecture Notes in Computer Science
Gaspers and M. Liedloff, A branch-and-reduce algorithm for finding a minimum independent dominating set in graphs, Proc. WG'06).  ...  We first devise a branching algorithm that computes a minimum independent dominating set on any graph with running time O*(2^0.424n) and polynomial space. This improves the O*(2^0.441n) result by (S.  ...  For the degree of v, we use the notation : v ∈ H}. We use δ and ∆ to denote the minimum and maximum degree of G, respectively.  ...

### Factoring matrices with a tree-structured sparsity pattern

Alex Druinsky, Sivan Toledo
2011 Linear Algebra and its Applications
We show that if the columns of A are ordered using minimum degree on A + A * , then factoring A using a sparse LU with partial pivoting algorithm generates only O(d max n) fill, requires only O(d max n  ...  We also propose an even more efficient and just-as-stable algorithm called sibling-dominant pivoting.  ...  This research was supported by an IBM Faculty Partnership Award, by grants 848/04 and 1045/09 from the Israel Science Foundation (founded by the Israel Academy of Sciences and Humanities), and by grant  ...

### Approximation hardness of dominating set problems in bounded degree graphs

M. Chlebík, J. Chlebíková
2008 Information and Computation
We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs.  ...  degree graphs.  ...  Theorem 1 . 1 Minimum Dominating Set, Minimum Total Dominating Set, and Minimum Connected Dominating Set cannot be approximated to within a factor of (1 − ε) ln n in polynomial time for any constant ε  ...

### Distributed algorithms for edge dominating sets

Jukka Suomela
2010 Proceeding of the 29th ACM SIGACT-SIGOPS symposium on Principles of distributed computing - PODC '10
Edge-based covering problems in port-numbering model 14 • Minimum edge cover seems to be a bit too simple: factor 2 approximation is trivial and tight• But what about minimum edge dominating sets?  ...  • Contribution: full characterisation of approximability of edge dominating sets in regular graphs and bounded-degree graphs Edge dominating sets: deterministic algorithms in port-numbering model  ...

### Page 8949 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews
C. (1-CLEM; Clemson, SC) Factor domination and minimum degree. (English summary) Discrete Math. 262 (2003), no. 1-3, 113-119.  ...  We prove that in any graph of order n, minimum degree 6 and maximum degree A#0, IR(G) <n/(1+6/A) and IR(G) — B(G) < ((A—2)/2A)n. The two bounds are attained by arbitrarily large graphs.  ...

### On the number of components in 2-factors of claw-free graphs

Kiyoshi Yoshimoto
2007 Discrete Mathematics
Additionally, we give a family of claw-free graphs with minimum degree 4 in which every 2-factor contains more than n/ components.  ...  In this paper, we prove that if a claw-free graph G with minimum degree 4 has no maximal clique of two vertices, then G has a 2-factor with at most (|G| − 1)/4 components.  ...  The edge-degree of an edge uv is defined as d(u) + d(v) − 2 and the minimum edge-degree e (G) is the minimum number of the edge-degrees of all edges in G.  ...

### Page 1494 of Mathematical Reviews Vol. , Issue 2001C [page]

2001 Mathematical Reviews
A characterisation of connected graphs with minimum degree at least 2 and domination number exceeding a third their size is obtained.  ...  Connected graphs G of size q with minimum degree at least 2 satisfying y,(G) > q/2 are characterised.  ...

### Matching properties in connected domination critical graphs

Nawarat Ananchuen, Watcharaphong Ananchuen, Michael D. Plummer
2008 Discrete Mathematics
certain assumptions regarding connectivity and minimum degree, a critical graph G with (ordinary) domination number 3 will be factor-critical (if |V (G)| is odd), bicritical (if |V (G)| is even) or 3-  ...  Plummer, 3-factor-criticality in domination critical graphs, Discrete Math. 2007, to appear [3].] on ordinary (i.e., not necessarily connected) domination, the first and third authors showed that under  ...  Note that the minimum degree of the graph H 2n−6,2,2,2 is 3 for any n 4. 3-c-criticality and factor-criticality In the case of odd graphs, the minimum degree requirement necessary to guarantee factor-criticality  ...

### On the Complexity of Making a Distinguished Vertex Minimum or Maximum Degree by Vertex Deletion [article]

Sounaka Mishra, Ashwin Pananjady, N Safina Devi
2014 arXiv   pre-print
Given a vertex weighted graph G=(V,E) and a specified, or "distinguished" vertex p ∈ V, MDD(min) is the problem of finding a minimum weight vertex set S ⊆ V∖{p} such that p becomes the minimum degree vertex  ...  in G[V ∖ S]; and MDD(max) is the problem of finding a minimum weight vertex set S ⊆ V∖{p} such that p becomes the maximum degree vertex in G[V ∖ S].  ...  S ⊆ V is a dominating set in G if and only if p is the vertex of minimum degree in H[V ′ \ S] (i.e. S is a solution to MDD(min) for H). Proof : Let S ⊆ V be a dominating set in G.  ...
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