Connected dominating set

Last updated

In graph theory, a connected dominating set and a maximum leaf spanning tree are two closely related structures defined on an undirected graph.

Contents

Definitions

A connected dominating set of a graph G is a set D of vertices with two properties:

  1. Any node in D can reach any other node in D by a path that stays entirely within D. That is, D induces a connected subgraph of G.
  2. Every vertex in G either belongs to D or is adjacent to a vertex in D. That is, D is a dominating set of G.

A minimum connected dominating set of a graph G is a connected dominating set with the smallest possible cardinality among all connected dominating sets of G. The connected domination number of G is the number of vertices in the minimum connected dominating set. [1]

Any spanning tree T of a graph G has at least two leaves, vertices that have only one edge of T incident to them. A maximum leaf spanning tree is a spanning tree that has the largest possible number of leaves among all spanning trees of G. The max leaf number of G is the number of leaves in the maximum leaf spanning tree. [2]

Complementarity

If d is the connected domination number of an n-vertex graph G, where n > 2, and l is its max leaf number, then the three quantities d, l, and n obey the simple equation

[3]

If D is a connected dominating set, then there exists a spanning tree in G whose leaves include all vertices that are not in D: form a spanning tree of the subgraph induced by D, together with edges connecting each remaining vertex v that is not in D to a neighbor of v in D. This shows that lnd.

In the other direction, if T is any spanning tree in G, then the vertices of T that are not leaves form a connected dominating set of G. This shows that nld. Putting these two inequalities together proves the equality n = d + l.

Therefore, in any graph, the sum of the connected domination number and the max leaf number equals the total number of vertices. Computationally, this implies that determining the connected domination number is equally difficult as finding the max leaf number.

Algorithms

It is NP-complete to test whether there exists a connected dominating set with size less than a given threshold, or equivalently to test whether there exists a spanning tree with at least a given number of leaves. Therefore, it is believed that the minimum connected dominating set problem and the maximum leaf spanning tree problem cannot be solved in polynomial time.

When viewed in terms of approximation algorithms, connected domination and maximum leaf spanning trees are not the same: approximating one to within a given approximation ratio is not the same as approximating the other to the same ratio. There exists an approximation for the minimum connected dominating set that achieves a factor of 2 ln Δ + O(1), where Δ is the maximum degree of a vertex in G. [4] The maximum leaf spanning tree problem is MAX-SNP hard, implying that no polynomial time approximation scheme is likely. [5] However, it can be approximated to within a factor of 2 in polynomial time. [6]

Both problems may be solved, on n-vertex graphs, in time O(1.9n). [7] The maximum leaf problem is fixed-parameter tractable, meaning that it can be solved in time exponential in the number of leaves but only polynomial in the input graph size. The klam value of these algorithms (intuitively, a number of leaves up to which the problem can be solved within a reasonable amount of time) has gradually increased, as algorithms for the problem have improved, to approximately 37, [8] and it has been suggested that at least 50 should be achievable. [9]

In graphs of maximum degree three, the connected dominating set and its complementary maximum leaf spanning tree problem can be solved in polynomial time, by transforming them into an instance of the matroid parity problem for linear matroids. [10]

Applications

Connected dominating sets are useful in the computation of routing for mobile ad hoc networks. In this application, a small connected dominating set is used as a backbone for communications, and nodes that are not in this set communicate by passing messages through neighbors that are in the set. [11]

The max leaf number has been employed in the development of fixed-parameter tractable algorithms: several NP-hard optimization problems may be solved in polynomial time for graphs of bounded max leaf number. [2]

See also

Related Research Articles

<span class="mw-page-title-main">Spanning tree</span> Tree which includes all vertices of a graph

In the mathematical field of graph theory, a spanning treeT of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree. If all of the edges of G are also edges of a spanning tree T of G, then G is a tree and is identical to T.

<span class="mw-page-title-main">Steiner tree problem</span> On short connecting networks with added vertices

In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a number of settings, they all require an optimal interconnect for a given set of objects and a predefined objective function. One well-known variant, which is often used synonymously with the term Steiner tree problem, is the Steiner tree problem in graphs. Given an undirected graph with non-negative edge weights and a subset of vertices, usually referred to as terminals, the Steiner tree problem in graphs requires a tree of minimum weight that contains all terminals and minimizes the total weight of its edges. Further well-known variants are the Euclidean Steiner tree problem and the rectilinear minimum Steiner tree problem.

<span class="mw-page-title-main">Independent set (graph theory)</span> Unrelated vertices in graphs

In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set of vertices such that for every two vertices in , there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in . A set is independent if and only if it is a clique in the graph's complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening.

In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a network flow problem.

<i>k</i>-minimum spanning tree

The k-minimum spanning tree problem, studied in theoretical computer science, asks for a tree of minimum cost that has exactly k vertices and forms a subgraph of a larger graph. It is also called the k-MST or edge-weighted k-cardinality tree. Finding this tree is NP-hard, but it can be approximated to within a constant approximation ratio in polynomial time.

In graph theory, the metric dimension of a graph G is the minimum cardinality of a subset S of vertices such that all other vertices are uniquely determined by their distances to the vertices in S. Finding the metric dimension of a graph is an NP-hard problem; the decision version, determining whether the metric dimension is less than a given value, is NP-complete.

<span class="mw-page-title-main">Dominating set</span> Subset of a graphs nodes such that all other nodes link to at least one

In graph theory, a dominating set for a graph G is a subset D of its vertices, such that any vertex of G is either in D, or has a neighbor in D. The domination numberγ(G) is the number of vertices in a smallest dominating set for G.

In graph theory, a domatic partition of a graph is a partition of into disjoint sets , ,..., such that each Vi is a dominating set for G. The figure on the right shows a domatic partition of a graph; here the dominating set consists of the yellow vertices, consists of the green vertices, and consists of the blue vertices.

In the mathematical discipline of graph theory, a feedback vertex set (FVS) of a graph is a set of vertices whose removal leaves a graph without cycles. Equivalently, each FVS contains at least one vertex of any cycle in the graph. The feedback vertex set number of a graph is the size of a smallest feedback vertex set. The minimum feedback vertex set problem is an NP-complete problem; it was among the first problems shown to be NP-complete. It has wide applications in operating systems, database systems, and VLSI chip design.

<span class="mw-page-title-main">Feedback arc set</span> Edges that hit all cycles in a graph

In graph theory and graph algorithms, a feedback arc set or feedback edge set in a directed graph is a subset of the edges of the graph that contains at least one edge out of every cycle in the graph. Removing these edges from the graph breaks all of the cycles, producing an acyclic subgraph of the given graph, often called a directed acyclic graph. A feedback arc set with the fewest possible edges is a minimum feedback arc set and its removal leaves a maximum acyclic subgraph; weighted versions of these optimization problems are also used. If a feedback arc set is minimal, meaning that removing any edge from it produces a subset that is not a feedback arc set, then it has an additional property: reversing all of its edges, rather than removing them, produces a directed acyclic graph.

In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. These edges are said to cross the cut. In a connected graph, each cut-set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions.

The arboricity of an undirected graph is the minimum number of forests into which its edges can be partitioned. Equivalently it is the minimum number of spanning forests needed to cover all the edges of the graph. The Nash-Williams theorem provides necessary and sufficient conditions for when a graph is k-arboric.

In graph theory, a minimum degree spanning tree is a subset of the edges of a connected graph that connects all the vertices together, without any cycles, and its maximum degree of its vertices as small as possible. That is, it is a spanning tree whose maximum degree is minimal.

<span class="mw-page-title-main">Claw-free graph</span> Graph without four-vertex star subgraphs

In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph.

In graph theory, a clique cover or partition into cliques of a given undirected graph is a partition of the vertices into cliques, subsets of vertices within which every two vertices are adjacent. A minimum clique cover is a clique cover that uses as few cliques as possible. The minimum k for which a clique cover exists is called the clique cover number of the given graph.

<span class="mw-page-title-main">Maximum cut</span> Problem of finding a maximum cut in a graph

For a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets S and T, such that the number of edges between S and T is as large as possible. Finding such a cut is known as the max-cut problem.

Bidimensionality theory characterizes a broad range of graph problems (bidimensional) that admit efficient approximate, fixed-parameter or kernel solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, bounded-genus graphs and graphs excluding any fixed minor. In particular, bidimensionality theory builds on the graph minor theory of Robertson and Seymour by extending the mathematical results and building new algorithmic tools. The theory was introduced in the work of Demaine, Fomin, Hajiaghayi, and Thilikos, for which the authors received the Nerode Prize in 2015.

In network theory, the Wiener connector is a means of maximizing efficiency in connecting specified "query vertices" in a network. Given a connected, undirected graph and a set of query vertices in a graph, the minimum Wiener connector is an induced subgraph that connects the query vertices and minimizes the sum of shortest path distances among all pairs of vertices in the subgraph. In combinatorial optimization, the minimum Wiener connector problem is the problem of finding the minimum Wiener connector. It can be thought of as a version of the classic Steiner tree problem, where instead of minimizing the size of the tree, the objective is to minimize the distances in the subgraph.

<span class="mw-page-title-main">Matroid parity problem</span> Largest independent set of paired elements

In combinatorial optimization, the matroid parity problem is a problem of finding the largest independent set of paired elements in a matroid. The problem was formulated by Lawler (1976) as a common generalization of graph matching and matroid intersection. It is also known as polymatroid matching, or the matchoid problem.

<span class="mw-page-title-main">Cutwidth</span> Property in graph theory

In graph theory, the cutwidth of an undirected graph is the smallest integer with the following property: there is an ordering of the vertices of the graph, such that every cut obtained by partitioning the vertices into earlier and later subsets of the ordering is crossed by at most edges. That is, if the vertices are numbered , then for every , the number of edges with and is at most .

References

  1. Sampathkumar, E.; Walikar, HB (1979), "The connected domination number of a graph", J. Math. Phys. Sci, 13 (6): 607–613.
  2. 1 2 Fellows, Michael; Lokshtanov, Daniel; Misra, Neeldhara; Mnich, Matthias; Rosamond, Frances; Saurabh, Saket (2009), "The complexity ecology of parameters: an illustration using bounded max leaf number", Theory of Computing Systems, 45 (4): 822–848, doi:10.1007/s00224-009-9167-9, S2CID   4053586 .
  3. Douglas, Robert J. (1992), "NP-completeness and degree restricted spanning trees", Discrete Mathematics, 105 (1–3): 41–47, doi: 10.1016/0012-365X(92)90130-8 .
  4. Guha, S.; Khuller, S. (1998), "Approximation algorithms for connected dominating sets", Algorithmica, 20 (4): 374–387, doi:10.1007/PL00009201, hdl: 1903/830 , S2CID   263230631 .
  5. Galbiati, G.; Maffioli, F.; Morzenti, A. (1994), "A short note on the approximability of the maximum leaves spanning tree problem", Information Processing Letters, 52 (1): 45–49, doi:10.1016/0020-0190(94)90139-2 .
  6. Solis-Oba, Roberto (1998), "2-approximation algorithm for finding a spanning tree with maximum number of leaves", Proc. 6th European Symposium on Algorithms (ESA'98) , Lecture Notes in Computer Science, vol. 1461, Springer-Verlag, pp. 441–452, doi:10.1007/3-540-68530-8_37, hdl: 11858/00-001M-0000-0014-7BD6-0 .
  7. Fernau, Henning; Kneis, Joachim; Kratsch, Dieter; Langer, Alexander; Liedloff, Mathieu; Raible, Daniel; Rossmanith, Peter (2011), "An exact algorithm for the maximum leaf spanning tree problem", Theoretical Computer Science, 412 (45): 6290–6302, doi: 10.1016/j.tcs.2011.07.011 , MR   2883043 .
  8. Binkele-Raible, Daniel; Fernau, Henning (2014), "A parameterized measure-and-conquer analysis for finding a k-leaf spanning tree in an undirected graph", Discrete Mathematics & Theoretical Computer Science, 16 (1): 179–200, MR   3188035 .
  9. Fellows, Michael R.; McCartin, Catherine; Rosamond, Frances A.; Stege, Ulrike (2000), "Coordinatized kernels and catalytic reductions: an improved FPT algorithm for max leaf spanning tree and other problems", FST-TCS 2000: Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Comput. Sci., vol. 1974, Springer, Berlin, pp. 240–251, doi:10.1007/3-540-44450-5_19, MR   1850108 .
  10. Ueno, Shuichi; Kajitani, Yoji; Gotoh, Shin'ya (1988), "On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three", Proceedings of the First Japan Conference on Graph Theory and Applications (Hakone, 1986), Discrete Mathematics , 72 (1–3): 355–360, doi: 10.1016/0012-365X(88)90226-9 , MR   0975556
  11. Wu, J.; Li, H. (1999), "On calculating connected dominating set for efficient routing in ad hoc wireless networks", Proceedings of the 3rd International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, ACM, pp. 7–14, doi:10.1145/313239.313261, S2CID   59969437 .