William Chung-Kung Yen

The Hub Number of Sierpiński-Like Graphs

A set Q⊆V is a hub set of a graph G=(V,E) if, for every pair of vertices u,v∈V∖Q, there exists a path from u to v such that all intermediate vertices are in Q. The hub number of G is the minimum size of a hub set in G. This paper derives the hub numbers of Sierpiński-like graphs including: Sierpiński graphs, extended Sierpiński graphs, and Sierpiński gasket graphs. Meanwhile, the corresponding minimum hub sets are also obtained.

The connected p-center problem on block graphs with forbidden vertices

Let G(V,E,w,l) denote an n-vertex and m-edge graph in which w is a function mapping each vertex v to a positive weight w(v) and l is a function mapping each edge e to a positive length l(e). Given a positive integer p, the p-Center problem involves finding a set Q with p vertices of G to be the locations for building facilities. The objective is to minimize the maximum weighted distance from each vertex in V-Q to its nearest vertex in Q. This paper considers a practical restriction: the induced subgraph of the selected p vertices must be connected. The new variant is called the Connected p-Center problem (the CpC problem). For each fixed integer t>=1, on block graphs with exactly t blocks, we first show that the CpC problem is NP-hard when (1) w(v)=1, for all vertices v, and l(e)@?{1,2}, for all edges e, and (2) w(v)@?{1,2}, for all vertices v, and l(e)=1, for all edges e, respectively. Second, an O(n+m)-time algorithm for solving the CpC problem on block graphs with unit vertex-weights and unit edge-lengths is proposed. Then, the algorithmic result is extended to handle the situation in which some vertices in G cannot be included to form feasible solutions. The complexity of the extended algorithm is also O(n+m).

Domatic Partition on Several Classes of Graphs

The domatic number of a graph G, denoted by DN(G), is the maximum number k such that V can be partitioned into k disjoint dominating sets. The domatic partition problem is to find a partition of the vertices of G into DN(G) dominating sets. The k-domatic partition problem with fixed k is to find a partition of the vertices of G into k dominating sets. In this paper, we show that 3-domatic partition problem is NP-complete on planar bipartite graphs, and the domatic partition problem is NP-complete on co-bipartite graphs. We further show that the unique 3-domatic partition problem is NP-hard on general graphs. Moreover, we propose an O(n)-time algorithm on the 3-domatic partition problem for maximal planar graphs, and O(n 3)-time algorithms on the domatic partition problem for P 4-sparse graphs and tree-cographs, respectively.

Weighted restrained domination in subclasses of planar graphs

Abstract An undirected simple and connected graph is denoted by G ( V , E ) , where V and E are the vertex-set and the edge-set of G , respectively. For any subset S of V , 〈 S 〉 denotes the subgraph of G induced by S . A subset Q of V is a restrained dominating set of G if ⋃ u ∈ Q N [ u ] = V and no isolated vertices appear in 〈 V − Q 〉 , where N ( x ) = { u | ( u , x ) ∈ E } and N [ x ] = N ( x ) ∪ { x } , for all x ∈ V . This paper studies the Weighted Restrained Domination problem (the WRD problem) on graphs G ( V , E , w ) , where w is a function giving a positive weight w ( v ) to each vertex v . The problem is to find a restrained dominating set D of the given graph G ( V , E , w ) and the objective is to minimize w ( D ) = ∑ v ∈ D w ( v ) . When w ( v ) = 1 , for all v ∈ V , the WRD problem is abbreviated as the RD problem. The first result shows that the WRD problem on cactus graphs can be solved in linear time. The second result proves that the RD problem on planar bipartite graphs remains NP-hard.

The connected p -median problem on block graphs

In this paper, we study a variant of the p-median problem on block graphs G in which the p-median is asked to be connected, and this problem is called the connected p-median problem. We first show that the connected p-median problem is NP-hard on block graphs with multiple edge weights. Then, we propose an O(n)-time algorithm for solving the problem on unit-edge-weighted block graphs, where n is the number of vertices in G.

The hub number of co-comparability graphs

A set H ? V is a hub set of a graph G = ( V , E ) if, for every pair of vertices u , v ? V ? H , either u is adjacent to v or there exists a path from u to v such that all intermediate vertices are in H. The hub number of G, denoted by h ( G ) , is the minimum size of a hub set in G. The connected hub number of G, denoted by h c ( G ) , is the minimum size of a connected hub set in G. In this paper, we prove that h ( G ) = h c ( G ) for co-comparability graphs G and characterize the case for which γ c ( G ) = h c ( G ) in this class of graphs, where γ c ( G ) denotes the connected domination number of G. We also show that h ( G ) can be computed in O ( | V | ) time for trapezoid graphs and in O ( | V | 3 ) time for co-comparability graphs.

An Optimal Algorithm for Untangling Binary Trees via Rotations

There are various ways to measure the shape difference between two n-node rooted binary trees (binary trees for short). A rotation on a binary tree is a local restructuring that changes the tree into another one preserving the in-order sequence. The rotation distance between two binary trees is the minimum number of rotations needed to transform one into another. Till now, no polynomial–time algorithm exists for computing the rotation distance between any two binary trees. Recently, Lucas (Comput. J., 47, 259–269, 2004) presented an O(n2)–time algorithm for finding the rotation distance between two binary trees, where the source tree is a degenerate tree and the destination tree is an angle tree. This paper improves the time-complexity to O(n) under this constraint.

Edge-Orienting on Split, Planar and Treelike Graphs

Let G(V, E) be an undirected connected graph, where each vertex v is associated with a positive cost C(v) and each edge e 5 (u, v) is associated with two positive weights, W(u! v) and W(v! u). We consider a new graph problem, called the edge-orientation problem (the EOP). The major issue is to assign each edge e 5 (u, v) an orientation, either from u to v, denoted as u! v, or from v to u, denoted as v! u, such that maxx[VfC(x) 1 Sx!z W(x! z)g is minimized. This paper first shows that the EOP is NP-hard on split graphs and planar graphs. Then, a linear-time algorithm on star graphs is proposed by the prune-and-search strategy. Finally, the algorithmic result on star graphs is extended to trees and simple cactus graphs using the dynamic programming strategy.