Tree Convex Research Articles

Counting dominating sets in some subclasses of bipartite graphs

Counting dominating sets in a graph is a #P-complete problem even for bipartite graphs. This paper concerns the class of tree convex bipartite graphs, which is a subclass of bipartite graphs. A bipartite graph G with bipartition (X,Y) is called tree convex, if a tree T defined on X exists such that for every vertex y in Y, the neighbors of y form a subtree in T. For each y in Y, when the neighbors of y form a path or a rooted directed path in T, the tree convex bipartite graph is called path-tree bipartite or rooted path-tree bipartite, respectively. This work proves that the problem of counting dominating sets remains #P-complete for path-tree bipartite graphs but admits a polynomial-time solution for the more tightly constrained subclass of rooted path-tree bipartite graphs.

Counting independent sets and maximal independent sets in some subclasses of bipartite graphs

The problems of counting independent sets and maximal independent sets are #P-complete for tree convex bipartite graphs but solvable in polynomial time for its subclass of convex bipartite graphs. This study investigates these problems for so-called path–treebipartite graphs, which are a subclass of bipartite graphs between tree convex bipartite graphs and convex bipartite graphs. A bipartite graph G with bipartition (X, Y) is called a path-tree bipartite graph, if a tree T that is defined on X exists such that, for all vertices y in Y, the neighbors of y form a path in T. This study reveals that the problems of counting independent sets and maximal independent sets remain #P-complete even for path-tree bipartite graphs but a stricter restriction to rooted path–tree bipartite graphs admits polynomial-time solutions.

Hardness and Structural Results for Half-Squares of Restricted Tree Convex Bipartite Graphs

Let $$B=(X,Y,E)$$ be a bipartite graph. A half-square of B has one color class of B as vertex set, say X; two vertices are adjacent whenever they have a common neighbor in Y. Every planar graph is a half-square of a planar bipartite graph, namely of its subdivision. Until recently, only half-squares of planar bipartite graphs, also known as map graphs (Chen et al., in: Proceedings of the thirtieth annual ACM symposium on the theory of computing, STOC ’98, pp 514–523. https://doi.org/10.1145/276698.276865 , 1998; J ACM 49(2):127–138. https://doi.org/10.1145/506147.506148 , 2002), have been investigated, and the most discussed problem is whether it is possible to recognize these graphs faster and simpler than Thorup’s $$O(n^{120})$$ -time algorithm (Thorup, in: Proceedings of the 39th IEEE symposium on foundations of computer science (FOCS), pp 396–405. https://doi.org/10.1109/SFCS.1998.743490 , 1998). In this paper, we identify the first hardness case, namely that deciding if a graph is a half-square of a balanced bisplit graph is NP-complete. (Balanced bisplit graphs form a proper subclass of star convex bipartite graphs). For classical subclasses of tree convex bipartite graphs such as biconvex, convex, and chordal bipartite graphs, we give good structural characterizations of their half-squares that imply efficient recognition algorithms. As a by-product, we obtain new characterizations of unit interval graphs, interval graphs, and of strongly chordal graphs in terms of half-squares of biconvex bipartite, convex bipartite, and of chordal bipartite graphs, respectively. Good characterizations of half-squares of star convex and star biconvex bipartite graphs are also given, giving linear-time recognition algorithms for these half-squares.

On tree-preserving constraints

The study of tractable subclasses of constraint satisfaction problems is a central topic in constraint solving. Tree convex constraints are extensions of the well-known row convex constraints. Just like the latter, every path-consistent tree convex constraint network is globally consistent. However, it is NP-complete to decide whether a tree convex constraint network has solutions. This paper studies and compares three subclasses of tree convex constraints, which are called chain-, path-, and tree-preserving constraints respectively. The class of tree-preserving constraints strictly contains the subclasses of path-preserving and arc-consistent chain-preserving constraints. We prove that, when enforcing strong path-consistency on a tree-preserving constraint network, in each step, the network remains tree-preserving. This ensures the global consistency of consistent tree-preserving networks after enforcing strong path-consistency, and also guarantees the applicability of the partial path-consistency algorithms to tree-preserving constraint networks, which is usually much more efficient than the path-consistency algorithms for large sparse constraint networks. As an application, we show that the class of tree-preserving constraints is useful in solving the scene labelling problem.

Automated efficient computation of crown transparency from tree silhouette images

The transparency of trees is the most important indicator for a forest health assessment. This paper presents an efficient method for calculating the crown transparency coefficient from tree binary images. This coefficient is based on the automated quantification of the deep indentation, macro-hole and micro-hole densities. Circular structuring elements are introduced, among other things, to automatically find the significant biological size. The symmetric tree convex hull and the tree smoothed contour are defined to delineate the reference areas necessary to evaluate the above-mentioned densities. Statistical thresholds are proposed to eliminate human operator subjectivity, especially in the automated identification of anatomical elements such as soft and deep crown-indentations or macro and micro crown-holes. A point-wise transparency map is produced to better appreciate the origin of the visible skylight areas in the crown. The crown micro-hole density is calculated from the 0.1-to-0.5 transparency points, the crown macro-hole density from the 0.5-to-1 transparency points. We finally opt for weighting of the above three densities with regard to the importance of the symptoms they describe for a more relevant crown transparency coefficient. A comparative study on several trees from full-size and half-size binary images showed that our method is similar overall to the DSO and less sensitive to scale reduction.

Counting independent sets in tree convex bipartite graphs

The problems of counting independent sets, maximal independent sets, and independent perfect dominating sets are #P-complete for bipartite graphs, but can be solved in polynomial time for convex bipartite graphs, which are a subclass of bipartite graphs This paper studies these problems for tree convex bipartite graphs, which are a class of graphs between bipartite graphs and convex bipartite graphs. A bipartite graph G with bipartition (X Y) is called tree convex, if a tree T defined on X exists, such that for every vertex y in Y, the neighbors of y form a subtree of T If the associated tree T is simply a path, then G is just a convex bipartite graph. This paper first proves that the problems of counting independent sets, maximal independent sets, and independent perfect dominating sets remain #P-complete for tree convex bipartite graphs even when the associated tree T is only a comb or a star. This paper then presents polynomial-time algorithms to solve these problems for tree convex bipartite graphs when the associated tree T is restricted to a triad, which consists of three paths with one common endpoint.

Complexity of domination, hamiltonicity and treewidth for tree convex bipartite graphs

Tree convex bipartite graphs generalize convex bipartite graphs by associating a tree, instead of a path, with one set of the vertices, such that for every vertex in another set, the neighborhood o...

Feedback vertex sets on restricted bipartite graphs

A feedback vertex set (FVS) in a graph is a subset of vertices whose complement induces a forest. Finding a minimum FVS is NP-complete on bipartite graphs, but tractable on convex bipartite graphs and on chordal bipartite graphs. A bipartite graph is called tree convex, if a tree is defined on one part of the vertices, such that for every vertex in the other part, its neighborhood induces a subtree. When the tree is a path, a triad or a star, the bipartite graph is called convex bipartite, triad convex bipartite or star convex bipartite, respectively. We show that: (1) FVS is tractable on triad convex bipartite graphs; (2) FVS is NP-complete on star convex bipartite graphs and on tree convex bipartite graphs where the maximum degree of vertices on the tree is at most three.

A REVIEW OF TREE CONVEX SETS TEST

A collection of sets may have some interesting properties which help identify efficient algorithms for constraint satisfaction problems and combinatorial auction problems. One of the properties is tree convexity. A collection S of sets is tree convex if we can find a tree T whose nodes are the union of the sets of S and each set of S is the nodes of a subtree of T. This concept extends that of row convex sets each of which is an interval over a total ordering of the elements of the union of these sets. An interesting problem is to find efficient algorithms to test whether a collection of sets is tree convex. It is not known before if there exists a linear time algorithm for this test. In this paper, we review the materials that are the key to a linear algorithm: hypergraphs, a characterization of tree convex sets and the acyclic hypergraph test algorithm. Some typos in the original paper of the acyclicity test are corrected here. Some experiments show that the linear algorithm is significantly faster than a well‐known existing algorithm.

Properties of tree convex constraints

It is known that a tree convex network is globally consistent if it is path consistent. However, if a tree convex network is not path consistent, enforcing path consistency on it may not make it globally consistent. In this paper, we investigate the properties of some tree convex constraints under intersection and composition. As a result, we identify a sub-class of tree convex networks that are locally chain convex and strictly union closed. This class of problems can be made globally consistent by arc and path consistency and thus is tractable. Interestingly, we also find that some scene labeling problems can be modeled by tree convex constraints in a natural and meaningful way.

Set Intersection and Consistency in Constraint Networks

In this paper, we show that there is a close relation between consistency in a constraint network and set intersection. A proof schema is provided as a generic way to obtain consistency properties from properties on set intersection. This approach not only simplifies the understanding of and unifies many existing consistency results, but also directs the study of consistency to that of set intersection properties in many situations, as demonstrated by the results on the convexity and tightness of constraints in this paper. Specifically, we identify a new class of tree convex constraints where local consistency ensures global consistency. This generalizes row convex constraints. Various consistency results are also obtained on constraint networks where only some, in contrast to all in the existing work,constraints are tight.