The number of maximal independent sets in a connected graph
TL;DR: The maximum number of maximal independent sets which a connected graph on n vertices can have is determined, and the extremal graphs are completely characterize, thereby answering a question of Wilf.
About: This article is published in Discrete Mathematics.The article was published on 1988-02-01 and is currently open access. It has received 92 citations till now. The article focuses on the topics: Maximal independent set & Independent set.
Citations
More filters
TL;DR: A survey of results concerning algorithms, complexity, and applications of the maximum clique problem is presented and enumerative and exact algorithms, heuristics, and a variety of other proposed methods are discussed.
Abstract: The maximum clique problem is a classical problem in combinatorial optimization which finds important applications in different domains. In this paper we try to give a survey of results concerning algorithms, complexity, and applications of this problem, and also provide an updated bibliography. Of course, we build upon precursory works with similar goals [39, 232, 266].
1,065 citations
14 Jun 2009
TL;DR: This work incorporates domain knowledge about the composition of words that should have high or low probability in various topics using a novel Dirichlet Forest prior in a LatentDirichlet Allocation framework.
Abstract: Users of topic modeling methods often have knowledge about the composition of words that should have high or low probability in various topics. We incorporate such domain knowledge using a novel Dirichlet Forest prior in a Latent Dirichlet Allocation framework. The prior is a mixture of Dirichlet tree distributions with special structures. We present its construction, and inference via collapsed Gibbs sampling. Experiments on synthetic and real datasets demonstrate our model's ability to follow and generalize beyond user-specified domain knowledge.
436 citations
Cites background from "The number of maximal independent s..."
...Finally, we mention that although in the worst case the number of maximal cliques Q in a connected component of size |r| can grow exponentially as O(3) (Griggs et al., 1988), in our experiments Q is no larger than 3, due in part to Must-Linked words “collapsing” to single nodes in the Cannot-Link graph....
[...]
...…1 Cannot-Link (in this example 2). component of size |r| can grow exponentially as We discuss the encoding for this single connected com-O(3|r|/3)(Griggs et al., 1988), in our experimentsQ(r) ponent r now, deferring discussion of the complete en-is no larger than 3, due in part to…...
[...]
TL;DR: A simple graph-theoretical proof that the largest number of maximal independent vertex sets in a tree with n vertices is given by m( T), a result first proved by Wilf.
Abstract: We give a simple graph-theoretical proof that the largest number of maximal independent vertex sets in a tree with n vertices is given by \[ m( T ) = \begin{cases} 2^{k - 1} + 1& {\text{if }} n = 2k, \\ 2^k & {\text{if }} n = 2k + 1, \end{cases}\] a result first proved by Wilf [SIAM J. Algebraic Discrete Methods, 7 (1986), pp. 125–130]. We also characterize those trees achieving this maximum value. Finally we investigate some related problems.
73 citations
TL;DR: The number of maximum independent sets is shown to depend on the structure within the tree of the α-critical edges and the families of trees on which these maxima are achieved are given.
Abstract: A subset of vertices is a maximum independent set if no two of the vertices are joined by an edge and the subset has maximum cardinality. in this paper we answer a question posed by Herb Wilf. We show that the greatest number of maximum independent sets for a tree of n vertices is
We give the families of trees on which these maxima are achieved.
Proving which trees are extremal depends upon the structure of maximum independent sets in trees. This structure is described in terms of adjacency rules between three types of vertices, those which are in all, no, or some maximum independent sets. We show that vertices that are in some but not all maximum independent sets of the tree are joined in pairs by the α-critical edges (edges whose removal increases the size of a maximum independent set). The number of maximum independent sets is shown to depend on the structure within the tree of the α-critical edges.
67 citations
TL;DR: In this article, the problem of determining the largest number of maximum independent sets of a graph of order n is studied, and solutions to this problem are given for various classes of graphs, including general graphs, trees, forests, (connected) graphs with at most one cycle, connected graphs and triangle-free graphs.
Abstract: In this paper, we study the problem of determining the largest number of maximum independent sets of a graph of order n. Solutions to this problem are given for various classes of graphs, including general graphs, trees, forests, (connected) graphs with at most one cycle, connected graphs and triangle-free graphs. Extremal graphs achieving the maximum values are also given.
51 citations
Cites background from "The number of maximal independent s..."
...[8] gave the maximum value of mi(G) for a connected graph G of order n ≥ 6 and the extremal graphs achieving this value....
[...]
References
More filters
[...]
TL;DR: In this article, the maximum number of cliques possible in a graph with n nodes is determined and bounds are obtained for the number of different sizes of clique possible in such a graph.
Abstract: A clique is a maximal complete subgraph of a graph. The maximum number of cliques possible in a graph withn nodes is determined. Also, bounds are obtained for the number of different sizes of cliques possible in such a graph.
907 citations
TL;DR: In this paper, the maximal independent sets of vertices that any tree of n vertices can have were shown to have maximal number of maximal independent vertices, where vertices are independent sets.
Abstract: We find the largest number of maximal independent sets of vertices that any tree of n vertices can have.
104 citations
TL;DR: A theorem of Moon and Moser is generalized to determine the maximum number of maximal independent sets in a connected graph on n vertices for n sufficiently large, e.g., n > 50.
Abstract: Generalizing a theorem of Moon and Moser, we determine the maximum number of maximal independent sets in a connected graph on n vertices for n sufficiently large, eg, n > 50
102 citations