A PBN is well known as a mathematical model of complex network systems such as gene regulatory networks. In Boolean networks, interactions between nodes (e.g., genes) are modeled by Boolean functions. In PBNs, Boolean functions are switched probabilistically. In this paper, for a PBN, a simplified representation that is effective in analysis and control is proposed. First, after a polynomial representation of a PBN is briefly explained, a simplified representation is derived. Here, the steady-state value of the expected value of the state is focused, and is characterized by a minimum feedback vertex set of an interaction graph expressing interactions between nodes. Next, using this representation, input selection and stabilization are discussed. Finally, the proposed method is demonstrated by a biological example.

Irreversible k-conversion set is introduced in connection with the mathematical modeling of the spread of diseases or opinions. We show that the problem to find a minimum irreversible 2-conversion set can be solved in O(n2log 6n) time for graphs with maximum degree at most 3 (subcubic graphs) by reducing it to the graphic matroid parity problem, where n is the number of vertices in a graph. This affirmatively settles an open question posed by Kyncl et al. (2014).

A feedback vertex set F of an undirected graph G is a vertex subset of G whose removal results in a forest. Such a set F is said to be independent if F forms an independent set of G. In this paper, we study the problem of finding an independent feedback vertex set of a given graph with the minimum number of vertices, from the viewpoint of graph classes. This problem is NP-hard even for planar bipartite graphs of maximum degree four. However, we show that the problem is solvable in linear time for graphs having tree-like structures, more specifically, for bounded treewidth graphs, chordal graphs and cographs. We then give a fixed-parameter algorithm for planar graphs when parameterized by the solution size. Such a fixed-parameter algorithm is already known for general graphs, but our algorithm is exponentially faster than the known one.

We consider the minimum feedback vertex set problem for some bipartite graphs and degree-constrained graphs. We show that the problem is linear time solvable for bipartite permutation graphs and NP-hard for grid intersection graphs. We also show that the problem is solvable in O(n2log 6n) time for n-vertex graphs with maximum degree at most three.

In an undirected graph, the feedback vertex set (FVS for short) problem is to find a set of vertices of minimum cardinality whose removal makes the graph acyclic. The FVS has applications to several areas such that combinatorial circuit design, synchronous systems, computer systems, VLSI circuits and so on. The FVS problem is known to be NP-hard on general graphs but interesting polynomial solutions have been found for some special classes of graphs. In this paper, we present an O(n2.68 + γn) time algorithm for solving the FVS problem on trapezoid graphs, where γ is the total number of factors included in all maximal cliques.

In this paper, we study the feedback vertex set problem for trivalent Cayley graphs, and construct a minimum feedback vertex set in trivalent Cayley graphs using the result on cube-connected cycles and the Cayley graph representation of trivalent Cayley graphs.