The general σ all-ones problem for trees

  • Authors:

  • Xueliang Li;Chao Wang;Xiaoyan Zhang

  • Affiliations:

  • Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin 300071, PR China;College of Software, Nankai University, Tianjin 300071, PR China;School of Mathematics and Computer Science, Institute of Mathematics, Nanjing Normal University, Nanjing 210097, PR China

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2008

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Abstract

The general @s all-ones problem is defined as follows: Given a graph G=(V,E), where V and E denote the vertex-set and the edge-set of G, respectively. Denote C as an initial subset of V. The problem is to find a subset X of V such that for every vertex v of G@?C the number of vertices in X adjacent to v is odd while for every vertex v of C the number is even. X is called a solution to the problem. When C=@A, this problem is the so-called @s all-ones problem. If a vertex is viewed to be adjacent to itself, the @s all-ones problem is addressed as the @s^+ all-ones problem. The @s^+ all-ones problem has been studied extensively. However, the @s all-ones problem has received much less attention. Unlike the @s^+ all-ones problem, which has solutions for any graphs, the @s all-ones problem may not have solutions for many graphs, even for some very simple graphs like C"3 and P"5. So, it becomes an interesting question to find polynomial time algorithms to determine if for a given tree the problem has solutions. And if it does, to find a solution to the minimum @s all-ones problem. In this paper we present two algorithms of linear time to solve the general @s all-ones problem for trees. The first one is good for counting the number of solutions if solutions do exist, and the second one is good for solving the minimum @s all-ones problem. Furthermore, we can modify the algorithm slightly to solve the general minimum @s all-ones problem.