We commence the study of domination in the incidence graphs of combinatorial designs. Let @math be a combinatorial design and denote by @math the domination number of the incidence (Levy) graph of @math . We obtain a number of results about the domination numbers of various kinds of designs. For instance, a finite projective plane of order @math , which is a symmetric @math -design, has @math . We also show that for any symmetric @math -design it holds that @math . We study at depth the domination numbers of Steiner systems and in particular of Steiner triple systems. We show that a @math has @math and also obtain a number of upper bounds. The tantalizing conjecture that all Steiner triple systems on @math vertices have the same domination number is proposed and is verified up to @math . The structure of minimal dominating sets is also investigated, both for its own sake and as a tool in deriving lower bounds on @math . Finally, a number of open questions are proposed.
Laskar and Wallis in @cite_8 have obtained some results about the domination number of the line graph of @math , motivated by combinatorial chessboard problems. However, the domination number of @math itself was not considered by them. The recent thesis of H ' e ger @cite_4 is probably closeset in spirit to the present work: H ' e ger constructs dominating sets of a particular form in projective planes.
@cite_4 @cite_8
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The graph-theoretic study of combinatorial chessboard problems can be extended to the study of line graphs of graphs of combinatorial designs. In particular, the determination of optimal placements of rooks on a chessboard corresponds to the determination of domination parameters of graphs of block designs. The determination of one such parameter, the independence number, is shown to follow directly from classical results in design theory. Additionally, the domination number of graphs of finite projective planes is discussed. Introduction (S.T. Hedetniemi, R.C. Laskar). Theoretical. Chessboard Domination Problems (E.J. Cockayne). On the Queen Domination Problem (C.M. Grinstead, B. Hahne, D. Van Stone). Recent Problems and Results About Kernels in Directed Graphs (C. Berge, P. Duchet). Critical Concepts in Domination (D.P. Summer). The Bondage Number of a Graph (J.F. ). Chordal Graphs and Upper Irredundance, Upper Domination and Independence (M.S. Jacobson and K. Peters). Regular Totally Domatically Full Graphs (B. Zelinka). Domatically Critical and Domatically Full Graphs (D. Rall). On Generalised Minimal Domination Parameters for Paths (B. Bollobas, E.J. Cockayne, C.M. Mynhardt). New Models. Dominating Cliques in Graphs (M.B. Cozzens, L.L. Kelleher). Covering all Cliques of Graph (Z. Tuza). Factor Domination in Graphs (R.C. Brigham, R. Dutton). The Least Point Covering and Domination Numbers of a Graph (E. Sampathkumar). Algorithmic. Dominating Sets in Perfect Graphs (D.G. Corneil, L.K. Stewart). Unit Disk Graphs (B.N. Clark, C.J. Colbourn, D.S. Johnson). Permutation Graphs: Connected Domination and Steiner Trees (C.J. Colbourn, L.K. Stewart). The Discipline Number of a Graph (V. Chvatal, W.J. Cook). Best Location of Service Centers in a Tree-Like Network under Budget Constraints (J. McHugh, Y. Perl). Dominating Cycles in Halin Graphs (M. Skowronska, M.M. Syslo). Finding Dominating Cliques Efficiently, in Strongly Chordal Graphs and Undirected Path Graphs (D. Kratsch). On Minimum Dominating Sets with Minimum Intersection (D.L. Grinstead, P.J. Slater). Bibliography. Bibliography on Domination in Graphs and Some Basic Definitions of Domination Parameters (S.T. Hedetniemi, R.C. Laskar). A set L@?V(G) of a graph G=(V,E) is a liar's dominating set if (1) for all v@?V(G), |N"G[v]@?L|>=2 and (2) for every pair u,v@?V(G) of distinct vertices, |(N"G[u]@?N"G[v])@?L|>=3. A graph G=(V,E) admits a liar's dominating set if each of its connected component has at least three vertices. Given a graph G=(V,E) and an integer K, the liar's domination decision problem (LR-DOMDP) is to decide whether G has a liar's dominating set of cardinality at most K. Slater [P.J. Slater, Liar's Domination, Networks, 54(2) (2009) 70-74] proved that the LR-DOMDP is NP-complete for general graphs. Subsequently, Roden and Slater [M.L. Roden and P.J. Slater, Liar's domination in graphs, Discrete Math., 309(19) (2009) 5884-5890] showed a more general family of problems to each be NP-complete for bipartite graphs. Besides this result, no other algorithmic result for the liar's dominating set problem is available in the literature. In this paper, we first strengthen the complexity result of the LR-DOMDP by showing that this problem remains NP-complete for split graphs and hence for chordal graphs. Finally, we propose a linear time algorithm for computing a minimum cardinality liar's dominating set in a tree. Let G=(V,E) be a graph without isolated vertices and having at least 3 vertices. A set L@?V(G) is a liar@?s dominating set if (1) |N"G[v]@?L|>=2 for all v@?V(G), and (2) |(N"G[u]@?N"G[v])@?L|>=3 for every pair u,v@?V(G) of distinct vertices in G, where N"G[x]= y@?V|xy@?E @? x is the closed neighborhood of x in G. Given a graph G and a positive integer k, the liar@?s domination problem is to check whether G has a liar@?s dominating set of size at most k. The liar@?s domination problem is known to be NP-complete for general graphs. In this paper, we propose a linear time algorithm for computing a minimum cardinality liar@?s dominating set in a proper interval graph. We also strengthen the NP-completeness result of liar@?s domination problem for general graphs by proving that the problem remains NP-complete even for undirected path graphs which is a super class of proper interval graphs. In 1985, Fink and Jacobson gave a generalization of the concepts of domination and independence in graphs. For a positive integer k, a subset S of vertices in a graph G = (V, E) is k-dominating if every vertex of V − S is adjacent to at least k vertices in S. The subset S is k-independent if the maximum degree of the subgraph induced by the vertices of S is less or equal to k − 1. In this paper we survey results on k-domination and k-independence. Let Y be a subset of real numbers. A Y-dominating function of a graph G=(V,E) is a function f:V->Y such that @?"u"@?"N"""G"["v"]f(u)>=1 for all vertices [email protected]?V, where N"G[v]= v @? u|(u,v)@?E . Let f(S)[email protected]?"u"@?"Sf(u) for any subset S of V and let f(V) be the weight of f. The Y-domination problem is to find a Y-dominating function of minimum weight for a graph G=(V,E). In this paper, we study the variations of Y-domination such as k -domination, k-tuple domination, signed domination, and minus domination for some classes of graphs. We give formulas to compute the k -domination, k-tuple domination, signed domination, and minus domination numbers of paths, cycles, n-fans, n-wheels, n-pans, and n-suns. Besides, we present a unified approach to these four problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. This paper also gives complexity results for the problems on doubly chordal graphs, dually chordal graphs, bipartite planar graphs, chordal bipartite graphs, and planar graphs. A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3. A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number of vertices, @math the minimum degree, and @math the maximum degree. We show that every graph has a domatic partition with @math dominating sets and, moreover, that such a domatic partition can be found in polynomial-time. This implies a @math -approximation algorithm for domatic number, since the domatic number is always at most @math . We also show this to be essentially best possible. Namely, extending the approximation hardness of set cover by combining multiprover protocols with zero-knowledge techniques, we show that for every @math , a @math -approximation implies that @math . This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better. We also show that every graph has a domatic partition with @math dominating sets, where the "o(1)" term goes to zero as @math increases. This can be turned into an efficient algorithm that produces a domatic partition of @math sets. We initiate a study of the homomorphism domination exponent of a pair of graphs F and G, defined as the maximum real number c such that |Hom(F,T)|>=|Hom(G,T)|^c for every graph T. The problem of determining whether HDE(F,G)>=1 is known as the homomorphism domination problem, and its decidability is an important open question arising in the theory of relational databases. We investigate the combinatorial and computational properties of the homomorphism domination exponent, proving upper and lower bounds and isolating classes of graphs F and G for which HDE(F,G) is computable. In particular, we present a linear program computing HDE(F,G) in the special case, where F is chordal and G is series-parallel. Let @math be a 3-uniform hypergraph of order @math and size @math , and let @math be a subset of vertices of @math . The set @math is a strong transversal in @math if @math contains at least two vertices from every edge of @math . The strong transversal number @math of @math is the minimum size of a strong transversal in @math . We show that @math , and we characterize the hypergraphs that achieve equality in this bound. In particular, we show that the Fano plane is the only connected 3-uniform hypergraph @math of order @math and size @math that achieves equality in this bound. A set @math of vertices in a graph @math is a double total dominating set of @math if every vertex of @math is adjacent to at least two vertices in @math . The minimum cardinality of a double total dominating set of @math is the double total domination number @math of @math . Let @math be a connected graph of order @math with minimum degree at least three. As an application of our hypergraph results, we show that @math with equality if and only if @math is the Heawood graph (equivalently, the incidence bipartite graph of the Fano plane). Further if @math is not the Heawood graph, we show that @math , while if @math is a cubic graph different from the Heawood graph, we show that @math , and this bound is sharp. We say a vertex v in a graph Gcovers a vertex w if v=w or if v and w are adjacent. A subset of vertices of G is a dominating set if it collectively covers all vertices in the graph. The dominating set problem, which is NP-hard, consists of finding a smallest possible dominating set for a graph. The straightforward greedy strategy for finding a small dominating set in a graph consists of successively choosing vertices which cover the largest possible number of previously uncovered vertices. Several variations on this greedy heuristic are described and the results of extensive testing of these variations is presented. A more sophisticated procedure for choosing vertices, which takes into account the number of ways in which an uncovered vertex may be covered, appears to be the most successful of the algorithms which are analyzed. For our experimental testing, we used both random graphs and graphs constructed by test case generators which produce graphs with a given density and a specified size for the smallest dominating set. We found that these generators were able to produce challenging graphs for the algorithms, thus helping to discriminate among them, and allowing a greater variety of graphs to be used in the experiments. In this paper, we study the dominance relation under a stochastic setting. Let @math be a set of @math colored stochastic points in @math , each of which is associated with an existence probability. We investigate the problem of computing the probability that a realization of @math contains inter-color dominances, which we call the (CSD) problem. We propose the first algorithm to solve the CSD problem for @math in @math time. On the other hand, we prove that, for @math , even the CSD problem with a restricted color pattern is #P-hard. In addition, even if the existence probabilities are restricted to be @math , the problem remains #P-hard for @math . A simple FPRAS is then provided to approximate the desired probability in any dimension. We also study a variant of the CSD problem in which the dominance relation is considered with respect to not only the standard basis but any orthogonal basis of @math . Specifically, this variant, which we call the free-basis colored stochastic dominance (FBCSD) problem, considers the probability that a realization of @math contains inter-color dominances with respect to any orthogonal basis of @math . We show that the CSD problem is polynomial-time reducible to the FBCSD problem in the same dimension, which proves the #P-hardness of the latter for @math . Conversely, we reduce the FBCSD problem in @math to the CSD problem in @math , by which an @math time algorithm for the former is obtained. Finding a small dominating set is one of the most fundamental problems of traditional graph theory. In this paper, we present a new fully distributed approximation algorithm based on LP relaxation techniques. For an arbitrary parameter k and maximum degree Δ, our algorithm computes a dominating set of expected size O(kΔ2 k log Δ|DSOPT|) in O(k2) rounds where each node has to send O(k2Δ) messages of size O(logΔ). This is the first algorithm which achieves a non-trivial approximation ratio in a constant number of rounds. We study the influence of edge subdivision on the convex domination number. We show that in general an edge subdivision can arbitrarily increase and arbitrarily decrease the convex domination number. We also find some bounds for unicyclic graphs and we investigate graphs G for which the convex domination number changes after subdivision of any edge in G. The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(Δ)+2 and H(Δ)+2 are presented, where Δ is the maximum degree and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited or has at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (cn+1) n where cn ln k is the approximation factor for the node weighted Steiner tree problem (currently cn = 1.6103 ). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c+1) H(Δ) +c-1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644 ). Computation of the minimum dominating set problem is a classical discrete optimization problem in graph theory. Recently, it has found application in the network controlling theory. In this paper, there are compared three approaches of small dominating set computation. The first one is based on the integer linear programming. The second one is the combined hill-climbing algorithm which is based on the randomization and binary searching. Both are compared with a simply greedy algorithm. The experiments were conducted for three different graph classes. The integer linear programming achieved the best performance for the large real-world network's analysis. Let G be a graph and @t:V(G)->N be an assignment of thresholds to the vertices of G. A subset of vertices D is said to be dynamic monopoly (or simply dynamo) if the vertices of G can be partitioned into subsets D"0,D"1,...,D"k such that D"0=D and for any i=1,...,k-1 each vertex v in D"i"+"1 has at least t(v) neighbors in D"[email protected][email protected]?D"i. Dynamic monopolies are in fact modeling the irreversible spread of influence such as disease or belief in social networks. We denote the smallest size of any dynamic monopoly of G, with a given threshold assignment, by dyn(G). In this paper, we first define the concept of a resistant subgraph and show its relationship with dynamic monopolies. Then we obtain some lower and upper bounds for the smallest size of dynamic monopolies in graphs with different types of thresholds. Next we introduce dynamo-unbounded families of graphs and prove some related results. We also define the concept of a homogeneous society that is a graph with probabilistic thresholds satisfying some conditions and obtain a bound for the smallest size of its dynamos. Finally, we consider dynamic monopoly of line graphs and obtain some bounds for their sizes and determine the exact values in some special cases. The paper deals with a polling game on a graph. Initially, each vertex is colored white or black. At each round, each vertex is colored by the color shared by the majority of vertices in its neighborhood, at the previous round. (All recolorings are done simultaneously.) We say that a set W0 of vertices is a dynamic monopoly or dynamo if starting the game with the vertices of W0 colored white, the entire system is white after a finite number of rounds. D. Peleg (1998, Discrete Appl. Math.86, 262?273) asked how small a dynamic monopoly may be as a function of the number of vertices. We show that the answer is O(1). All graphs considered in this paper are finite graphs without loops and multiple edges. The domatic number of a graph was defined by E. J. Cockayne and S. T. Hedetniemi [1]. Later some related concepts were introduced. The same authors together with R. M. Dawes [2] have introduced the total domatic number; R. Laskar and S. T. Hedetniemi [3] have introduced the connected domatic number. A dominating set (or a total dominating set) in an undirected graph G is a subset D of the vertex set V(G) of G with the property that to each vertex x e V(G) — D (or to each vertex xeV(G) respectively) there exists a vertex y e D adjacent to x. A connected dominating set of G is a dominating set of G with the property that the subgraph of G induced by it is connected. A domatic (or total domatic, or connected domatic) partition of G is a partition of V(G), all of whose classes are dominating (or total dominating, or connected dominating, respectively) sets of G. The maximum number of classes of a domatic (or total domatic, or connected domatic) partition of G is called the domatic (or total domatic, or connected domatic, respectively) number of G. The domatic number of G is denoted by d(G), its total domatic number by <4(G), its connected domatic number by dc(G). The connected domatic number of a graph is well defined only for connected graphs; in a disconnected graph there exists no connected dominating set and thus no connected domatic partition, while in every connected graph there exists at least one connected domatic partition, namely that which consists of one class. The connected domatic number of G is closely related to the vertex connectivity number of G. If G is a connected graph, then a vertex cut of G is a subset R of V(G) with the property that the subgraph of G induced by V(G) — R is disconnected. If G is not a complete graph, then the vertex connectivity number x(G) is the minimum cardinality of a vertex cut of G. If G is a complete graph (i. e. without vertex cuts) with n vertices, then we put x(G) = n — 1. Lemma. Let G be a connected graph which is not complete, let R be its vertex cut, let D be its connected dominating set. Then DnR^0. A subset L ? V of a graph G = ( V , E ) is called a liar's dominating set of G if (i) | N G u ] ? L | ? 2 for every vertex u ? V , and (ii) | ( N G u ] ? N G v ] ) ? L | ? 3 for every pair of distinct vertices u , v ? V . The Min Liar Dom Set problem is to find a liar's dominating set of minimum cardinality of a given graph G and the Decide Liar Dom Set problem is the decision version of the Min Liar Dom Set problem. The Decide Liar Dom Set problem is known to be NP-complete for general graphs. In this paper, we first present approximation algorithms and hardness of approximation results of the Min Liar Dom Set problem in general graphs, bounded degree graphs, and p-claw free graphs. We then show that the Decide Liar Dom Set problem is NP-complete for doubly chordal graphs and propose a linear time algorithm for computing a minimum liar's dominating set in block graphs.