Drawing Planar Graphs Symmetrically, III: Oneconnected Planar Graphs

Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. This paper discusses symmetric drawings of oneconnected planar graphs. More specifically, we discuss planar (geometric) automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a planar drawing of G. Finding planar automorphisms is the first and most difficult step in constructing planar symmetric drawings of graphs. The problem of determining whether a given graph has a nontrivial geometric automorphism is NP-complete for general graphs. The two previous papers in this series have discussed the problem of drawing planar graphs with a maximum number of symmetries, for the restricted cases where the graph is triconnected and biconnected. This paper extends the previous results to cover planar graphs that are oneconnected. We present a linear time algorithm for drawing oneconnected planar graphs with a maximum number of symmetries.     

A preferential semantics for causal reasoning about action

One of the principal concerns in the research area of Reasoning about Action is determining the ramifications of actions in changing environments. A particular tendency emerging in recent literature endorses the explicit incorporation of causal knowledge in logic-based action theories. It is argued that causal extensions not only enhance the expressive power of theories of action, but may also provide more concise and intuitive representations. This paper investigates semantics for causal reasoning about action and change. It does so by exploring the role of several fundamental underlying principles, such as the Principle of Minimal Change and the Principle of Causal Change. This work culminates in a general unifying semantics for a class of action theories represented by a number of recent and influential approaches – in particular, the causal relationship approach of Thielscher and the causal systems with fixed-points suggested by McCain and Turner. The unifying augmented preferential semantics, emerging as a result of this study, captures both Principles of Change and shows their clear and distinct roles.     

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.     

Complexity results for structure-based causality

We give a precise picture of the computational complexity of causal relationships in Pearl's structural models, where we focus on causality between variables, event causality, and probabilistic causality. As for causality between variables, we consider the notions of causal irrelevance, cause, cause in a context, direct cause, and indirect cause. As for event causality, we analyze the complexity of the notions of necessary and possible cause, and of the sophisticated notions of weak and actual cause by Halpern and Pearl. In the course of this, we also prove an open conjecture by Halpern and Pearl, and establish other semantic results. We then analyze the complexity of the probabilistic notions of probabilistic causal irrelevance, likely causes of events, and occurrences of events despite other events. Moreover, we consider decision and optimization problems involving counterfactual formulas. To our knowledge, no complexity aspects of causal relationships in the structural-model approach have been considered so far, and our results shed light on this issue.     

On maximum induced matchings in bipartite graphs

The problem of finding a maximum induced matching is known to be NP-hard in general bipartite graphs. We strengthen this result by reducing the problem to some special classes of bipartite graphs such as bipartite graphs with maximum degree 3 or C₄-free bipartite graphs. On the other hand, we describe a new polynomially solvable case for the problem in bipartite graphs which deals with a generalization of bi-complement reducible graphs.     

On vertex ranking of a starlike graph

A vertex coloring c:V→{1,2,…,t} of a graph G=(V,E) is a vertex t-ranking if for any two vertices of the same color every path between them contains a vertex of larger color. The vertex ranking number χ_r(G) is the smallest value of t such that G has a vertex t-ranking. A χ_r(G)-ranking of G is said to be an optimal vertex ranking. In this paper, we present an O(|V|+|E|) time algorithm for finding an optimal vertex ranking of a starlike graph G=(V,E). Our result implies that an optimal vertex ranking of a split graph can be computed in linear time.     

关于图的局部调和着色

通过引入图的局部调和着色的概念,给出了任意图的局部调和着色的不可改进的上、下界;同时讨论了图的局部调和着色的一些性质,以及一些特殊图的局部调和着色数.     

无爪图成为可遍历的一个充分条件

本文给出下列定理:设G是阶为n≥3的连通无爪图,如果对每对不同的非邻顶点x,y有2|N(x)UN(y)|+d(x)+d(y)≥2n-5,则G是可遍历的.     

用键图模拟方法分析连杆助力器稳定性

用键图模拟车辆上广泛应用的连杆式助力器的工作过程,建立空间状态方程.并对状态变量的稳定性进行分析.找出提高助力器稳定性的措施.     

K_(1.3)—Free图成为哈米顿的一个邻域并条件

在本文中,我们给出下列定理:设G为阶是n≥3的2—连通,K_(13)—free图且满足NC(G)≥n—δ—2。则G为哈米顿的,这里NC(G)=min{|N(u)N(v)|E}。