Pak Ching Li

Variations of the maximum leaf spanning tree problem for bipartite graphs

The maximum leaf spanning tree problem is known to be NP-complete. In [M.S. Rahman, M. Kaykobad, Complexities of some interesting problems on spanning trees, Inform. Process. Lett. 94 (2005) 93-97], a variation on this problem was posed. This variation restricts the problem to bipartite graphs and asks, for a fixed integer K, whether or not the graph contains a spanning tree with at least K leaves in one of the partite sets. We show not only that this problem is NP-complete, but that it remains NP-complete for planar bipartite graphs of maximum degree 4. We also consider a generalization of a related decision problem, which is known to be polynomial-time solvable. We show the problem is still polynomial-time solvable when generalized to weighted graphs.

Splitting systems and separating systems

Abstract Suppose m and t are integers such that 0 (X, B ) where |X|=m, B is a set of ⌊m/2⌋ subsets of X, called blocks such that for every Y⊆X and |Y|=t, there exists a block B∈ B such that |B∩Y|=⌊t/2⌋ or |(X⧹B)∩Y|=⌊t/2⌋. We will give some results on splitting systems for t=2 or 4 which often depend on results from uniform separating systems. Suppose that m is an even integer, t 1 , t 2 are integers such that t1+t2⩽m. A uniform (m,t1,t2)-separating system is an ordered pair (X, B ) where |X|=m, B is a set of subsets of X of size m/2, called blocks, such that for every P⊆X, Q⊆X where |P|=t 1 , |Q|=t 2 and P∩Q=∅, there exists a block B∈ B for which either P⊆B, Q∩B=∅ or Q⊆B, P∩B=∅ . We also give new results for separating systems.

Ranking and loopless generation of k-ary dyck words in cool-lex order

A binary string B of length n=kt is a k-ary Dyck word if it contains t copies of 1, and the number of 0s in every prefix of B is at most k−1 times the number of 1s. We provide two loopless algorithms for generating k-ary Dyck words in cool-lex order: (1) The first requires two index variables and assumes k is a constant; (2) The second requires t index variables and works for any k. We also efficiently rank k-ary Dyck words in cool-lex order. Our results generalize the “coolCat” algorithm by Ruskey and Williams (Generating balanced parentheses and binary trees by prefix shifts in CATS 2008) and provide the first loopless and ranking applications of the general cool-lex Gray code by Ruskey, Sawada, and Williams (Binary bubble languages and cool-lex order under review).

Facilitating visual queries in the treemap using distortion techniques

TreeMap is one a common space-filling visualization technique to display large hierarchies in a limited display space. In TreeMaps, highlighting techniques are widely used to depict search results from visual queries. To improve visualizing the queries results in the TreeMap, we designed a continuous animated multi-distortion algorithm based on fisheye and continuous zooming techniques. To evaluate the effectiveness of the new algorithm, we conducted an experiment to compare the distortion technique to the traditional highlighting methods used in TreeMaps. The results suggest that the multi-distortion technique is only effective with small result sets but not as effective as simple highlighting for large search result sets.

Constructions and bounds for (m,t)-splitting systems

Let m and t be positive integers with t>=2. An (m,t)-splitting system is a pair (X,B) where |X|=m and B is a collection of subsets of X called blocks such that for every Y@?X with |Y|=t, there exists a block B@?B such that |B@?Y|=@?t/2@?. An (m,t)-splitting system is uniform if every block has size @?m/2@?. In this paper, we give several constructions and bounds for splitting systems, concentrating mainly on the case t=3. We consider uniform splitting systems as well as other splitting systems with special properties, including disjunct and regular splitting systems. Some of these systems have interesting connections with other types of set systems.

Cycle-maximal triangle-free graphs

We conjecture that the balanced complete bipartite graph K ? n / 2 ? , ? n / 2 ? contains more cycles than any other n -vertex triangle-free graph, and we make some progress toward proving this. We give equivalent conditions for cycle-maximal triangle-free graphs; show bounds on the numbers of cycles in graphs depending on numbers of vertices and edges, girth, and homomorphisms to small fixed graphs; and use the bounds to show that among regular graphs, the conjecture holds. We also consider graphs that are close to being regular, with the minimum and maximum degrees differing by at most a positive integer k . For k = 1 , we show that any such counterexamples have n ? 91 and are not homomorphic to C 5 ; and for any fixed k there exists a finite upper bound on the number of vertices in a counterexample. Finally, we describe an algorithm for efficiently computing the matrix permanent (a # P -complete problem in general) in a special case used by our bounds.