6,339 research outputs found

    An Efficient Algorithm for Mixed Domination on Generalized Series-Parallel Graphs

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    A mixed dominating set SS of a graph G=(V,E)G=(V,E) is a subset SVE S \subseteq V \cup E such that each element v(VE)Sv\in (V \cup E) \setminus S is adjacent or incident to at least one element in SS. The mixed domination number γm(G)\gamma_m(G) of a graph GG is the minimum cardinality among all mixed dominating sets in GG. The problem of finding γm(G)\gamma_{m}(G) is know to be NP-complete. In this paper, we present an explicit polynomial-time algorithm to construct a mixed dominating set of size γm(G)\gamma_{m}(G) by a parse tree when GG is a generalized series-parallel graph

    On Complexities of Minus Domination

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    A function f: V \rightarrow \{-1,0,1\} is a minus-domination function of a graph G=(V,E) if the values over the vertices in each closed neighborhood sum to a positive number. The weight of f is the sum of f(x) over all vertices x \in V. The minus-domination number \gamma^{-}(G) is the minimum weight over all minus-domination functions. The size of a minus domination is the number of vertices that are assigned 1. In this paper we show that the minus-domination problem is fixed-parameter tractable for d-degenerate graphs when parameterized by the size of the minus-dominating set and by d. The minus-domination problem is polynomial for graphs of bounded rankwidth and for strongly chordal graphs. It is NP-complete for splitgraphs. Unless P=NP there is no fixed-parameter algorithm for minus-domination. 79,1 5

    Power domination throttling

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    A power dominating set of a graph G=(V,E)G=(V,E) is a set SVS\subset V that colors every vertex of GG according to the following rules: in the first timestep, every vertex in N[S]N[S] becomes colored; in each subsequent timestep, every vertex which is the only non-colored neighbor of some colored vertex becomes colored. The power domination throttling number of GG is the minimum sum of the size of a power dominating set SS and the number of timesteps it takes SS to color the graph. In this paper, we determine the complexity of power domination throttling and give some tools for computing and bounding the power domination throttling number. Some of our results apply to very general variants of throttling and to other aspects of power domination.Comment: 19 page

    Layered graphs: a class that admits polynomial time solutions for some hard problems

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    The independent set on a graph G=(V,E)G=(V,E) is a subset of VV such that no two vertices in the subset have an edge between them. The MIS problem on GG seeks to identify an independent set with maximum cardinality, i.e. maximum independent set or MIS. VVV* \subseteq V is a vertex cover G=(V,E)G=(V,E) if every edge in the graph is incident upon at least one vertex in VV*. VVV* \subseteq V is dominating set of G=(V,E)G=(V,E) if forall vVv \in V either vVv \in V* or uV\exists u \in V* and (u,v)E(u,v) \in E. A connected dominating set, CDS, is a dominating set that forms a single component in GG. The MVC problem on GG seeks to identify a vertex cover with minimum cardinality, i.e. minimum vertex cover or MVC. Likewise, CVC seeks a connected vertex cover (CVC) with minimum cardinality. The problems MDS and CDS seek to identify a dominating set and a connected dominating set respectively of minimum cardinalities. MVC, CVC, MDS, and CDS on a general graph are known to be NP-complete. On certain classes of graphs they can be computed in polynomial time. Such algorithms are known for bipartite graphs, chordal graphs, cycle graphs, comparability graphs, claw-free graphs, interval graphs and circular arc graphs for some of these problems. In this article we introduce a new class of graphs called a layered graph and show that if the number of vertices in a layer is O(logV)O(\log \mid V \mid) then MIS, MVC, CVC, MDS and CDC can be computed in polynomial time. The restrictions that are employed on graph classes that admit polynomial time solutions for hard problems, e.g. lack of cycles, bipartiteness, planarity etc. are not applicable for this class. \\ Key words: Independent set, vertex cover, dominating set, dynamic programming, complexity, polynomial time algorithms.Comment: 14 pages, 1 figure. A generic algorithm is given. It can be extended to handle a wide range of hard problems. Space complexity was incorrectly given as O(k2)O(k^2) for MIS (identical for MVC) in the earlier version instead of O(k2k)O(k 2^k). The edges in LLGLLG are more clearly defined. For Vit V_{it} the only permissible edges are (Vit,Vjt)(V_{it}, V_{jt}) where $j \in \{ i-1, i+1\}

    A Linear Algorithm for Computing γ[1,2]\gamma_{[1,2]}-set in Generalized Series-Parallel Graphs

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    For a graph G=(V,E)G=(V,E), a set SVS \subseteq V is a [1,2][1,2]-set if it is a dominating set for GG and each vertex vVSv \in V \setminus S is dominated by at most two vertices of SS, i.e. 1N(v)S21 \leq \vert N(v) \cap S \vert \leq 2. Moreover a set SVS \subseteq V is a total [1,2][1,2]-set if for each vertex of VV, it is the case that 1N(v)S21 \leq \vert N(v) \cap S \vert \leq 2. The [1,2][1,2]-domination number of GG, denoted γ[1,2](G)\gamma_{[1,2]}(G),is the minimum number of vertices in a [1,2][1,2]-set. Every [1,2][1,2]-set with cardinality of γ[1,2](G)\gamma_{[1,2]}(G) is called a γ[1,2]\gamma_{[1,2]}-set. Total [1,2][1,2]-domination number and γt[1,2]\gamma_{t[1,2]}-sets of GG are defined in a similar way. This paper presents a linear time algorithm to find a γ[1,2]\gamma_{[1,2]}-set and a γt[1,2]\gamma_{t[1,2]}-set in generalized series-parallel graphs

    Weighted domination number of cactus graphs

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    In the paper, we write a linear algorithm for calculating the weighted domination number of a vertex-weighted cactus. The algorithm is based on the well known depth first search (DFS) structure. Our algorithm needs less than 12n+5b12n+5b additions and 9n+2b9n+2b min\min-operations where nn is the number of vertices and bb is the number of blocks in the cactus.Comment: 17 pages, figures, submitted to Discussiones Mathematicae Graph Theor

    Efficient Encoding of Watermark Numbers as Reducible Permutation Graphs

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    In a software watermarking environment, several graph theoretic watermark methods use numbers as watermark values, where some of these methods encode the watermark numbers as graph structures. In this paper we extended the class of error correcting graphs by proposing an efficient and easily implemented codec system for encoding watermark numbers as reducible permutation flow-graphs. More precisely, we first present an efficient algorithm which encodes a watermark number ww as self-inverting permutation π\pi^* and, then, an algorithm which encodes the self-inverting permutation π\pi^* as a reducible permutation flow-graph F[π]F[\pi^*] by exploiting domination relations on the elements of π\pi^* and using an efficient DAG representation of π\pi^*. The whole encoding process takes O(n) time and space, where nn is the binary size of the number ww or, equivalently, the number of elements of the permutation π\pi^*. We also propose efficient decoding algorithms which extract the number ww from the reducible permutation flow-graph F[π]F[\pi^*] within the same time and space complexity. The two main components of our proposed codec system, i.e., the self-inverting permutation π\pi^* and the reducible permutation graph F[π]F[\pi^*], incorporate important structural properties which make our system resilient to attacks

    Parameterized Complexity of Safe Set

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    In this paper we study the problem of finding a small safe set SS in a graph GG, i.e. a non-empty set of vertices such that no connected component of G[S]G[S] is adjacent to a larger component in GSG - S. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[2]-hard when parameterized by the pathwidth pwpw and cannot be solved in time no(pw)n^{o(pw)} unless the ETH is false, (2) it admits no polynomial kernel parameterized by the vertex cover number vcvc unless PH=Σ3p\mathrm{PH} = \Sigma^{\mathrm{p}}_{3}, but (3) it is fixed-parameter tractable (FPT) when parameterized by the neighborhood diversity ndnd, and (4) it can be solved in time nf(cw)n^{f(cw)} for some double exponential function ff where cwcw is the clique-width. We also present (5) a faster FPT algorithm when parameterized by solution size

    Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1-Sperner hypergraphs

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    A hypergraph is said to be 11-Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of 11-Sperner hypergraphs and their structure to graphs. In particular, we consider the classical characterizations of threshold and domishold graphs and use them to obtain further characterizations of these classes in terms of 11-Spernerness, thresholdness, and 22-asummability of their vertex cover, clique, dominating set, and closed neighborhood hypergraphs. Furthermore, we apply a decomposition property of 11-Sperner hypergraphs to derive decomposition theorems for two classes of split graphs, a class of bipartite graphs, and a class of cobipartite graphs. These decomposition theorems are based on certain matrix partitions of the corresponding graphs, giving rise to new classes of graphs of bounded clique-width and new polynomially solvable cases of several domination problems.Comment: 31 pages, 9 figure

    Algorithms for Unipolar and Generalized Split Graphs

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    A graph G=(V,E)G=(V,E) is a {\it unipolar graph} if there exits a partition V=V1V2V=V_1 \cup V_2 such that, V1V_1 is a clique and V2V_2 induces the disjoint union of cliques. The complement-closed class of {\it generalized split graphs} are those graphs GG such that either GG {\it or} the complement of GG is unipolar. Generalized split graphs are a large subclass of perfect graphs. In fact, it has been shown that almost all C5C_5-free (and hence, almost all perfect graphs) are generalized split graphs. In this paper we present a recognition algorithm for unipolar graphs that utilizes a minimal triangulation of the given graph, and produces a partition when one exists. Our algorithm has running time O(nmnm^\prime), where mm^\prime is the number of edges in a minimal triangulation of the given graph. Generalized split graphs can recognized via this algorithm in O(nm' + n\OL{m}') = O(n3n^3) time. We give algorithms on unipolar graphs for finding a maximum independent set and a minimum clique cover in O(n+mn+m) time and for finding a maximum clique and a minimum proper coloring in O(n2.5/lognn^{2.5}/\log n), when a unipolar partition is given. These algorithms yield algorithms for the four optimization problems on generalized split graphs that have the same worst-case time bound. We also prove that the perfect code problem is NP-Complete for unipolar graphs.Comment: Please cite this article in press as: E.M. Eschen, X. Wang, Algorithms for unipolar and generalized split graphs. Discrete Applied Mathematics (2013),http://dx.doi.org/10.1016/j.dam.2013.0801
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