The Firebreak Problem

Suppose we have a network that is represented by a graph $G$. Potentially a fire (or other type of contagion) might erupt at some vertex of $G$. We are able to respond to this outbreak by establishing a firebreak at $k$ other vertices of $G$, so that the fire cannot pass through these fortified vertices. The question that now arises is which $k$ vertices will result in the greatest number of vertices being saved from the fire, assuming that the fire will spread to every vertex that is not fully behind the $k$ vertices of the firebreak. This is the essence of the {\sc Firebreak} decision problem, which is the focus of this paper. We establish that the problem is intractable on the class of split graphs as well as on the class of bipartite graphs, but can be solved in linear time when restricted to graphs having constant-bounded treewidth, or in polynomial time when restricted to intersection graphs. We also consider some closely related problems

Global and Fixed-Terminal Cuts in Digraphs

The computational complexity of multicut-like problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed graphs: double cut and bicut. 1. Fixed-terminal edge-weighted double cut is known to be solvable efficiently. We show that fixed-terminal node-weighted double cut cannot be approximated to a factor smaller than 2 under the Unique Games Conjecture (UGC), and we also give a 2-approximation algorithm. For the global version of the problem, we prove an inapproximability bound of 3/2 under UGC. 2. Fixed-terminal edge-weighted bicut is known to have an approximability factor of 2 that is tight under UGC. We show that the global edge-weighted bicut is approximable to a factor strictly better than 2, and that the global node-weighted bicut cannot be approximated to a factor smaller than 3/2 under UGC. 3. In relation to these investigations, we also prove two results on undirected graphs which are of independent interest. First, we show NP-completeness and a tight inapproximability bound of 4/3 for the node-weighted 3-cut problem under UGC. Second, we show that for constant k, there exists an efficient algorithm to solve the minimum {s,t}-separating k-cut problem. Our techniques for the algorithms are combinatorial, based on LPs and based on the enumeration of approximate min-cuts. Our hardness results are based on combinatorial reductions and integrality gap instances

The Structure of Minimum Vertex Cuts

In this paper we continue a long line of work on representing the cut structure of graphs. We classify the types of minimum vertex cuts, and the possible relationships between multiple minimum vertex cuts. As a consequence of these investigations, we exhibit a simple O(? n)-space data structure that can quickly answer pairwise (?+1)-connectivity queries in a ?-connected graph. We also show how to compute the "closest" ?-cut to every vertex in near linear O?(m+poly(?)n) time

Approximating minimum cost connectivity problems

We survey approximation algorithms of connectivity problems. The survey presented describing various techniques. In the talk the following techniques and results are presented. 1)Outconnectivity: Its well known that there exists a polynomial time algorithm to solve the problems of finding an edge k-outconnected from r subgraph [EDMONDS] and a vertex k-outconnectivity subgraph from r [Frank-Tardos] . We show how to use this to obtain a ratio 2 approximation for the min cost edge k-connectivity problem. 2)The critical cycle theorem of Mader: We state a fundamental theorem of Mader and use it to provide a 1+(k-1)/n ratio approximation for the min cost vertex k-connected subgraph, in the metric case. We also show results for the min power vertex k-connected problem using this lemma. We show that the min power is equivalent to the min-cost case with respect to approximation. 3)Laminarity and uncrossing: We use the well known laminarity of a BFS solution and show a simple new proof due to Ravi et al for Jain\u27s 2 approximation for Steiner network

Effects of Light and Macroinvertebrate Consumers on Detrital Microbial Biofilms in Streams

In lotic freshwater systems, aquatic macroinvertebrates are key processors of biofilms that grow upon organic matter. Although macroinvertebrate effects on biofilms may depend on light availability, the combined effects of consumers and light remain unexplored. Here, I conducted experiments to test effects of presence/absence of the omnivorous shrimp Macrobrachium ohione and the shredding caddisfly Pycnopsyche sp. on Liriodendron tulipifera litter biofilms in experimental streams under light or darkness. I measured litter-associated algal, fungal and bacterial biomasses and production rates, as well as litter decomposition, over 49 days. Both experiments exhibited significant positive effects of light on algal productivity and interactions of Macrobrachium and Pycnopsyche presence with time and light. Light increased bacterial productivity in the Pycnopsyche experiment, but not in the Macrobrachium experiment, in which time, light, and Macrobrachium interactively affected bacterial production. Litter decomposition was unaffected by light or Macrobrachium presence, but Pycnopsyche presence increased decomposition rates. My results suggest that light strongly affects litter biofilms, whereas consumers primarily affect the timing and succession of periphytic microbial colonization of organic matter. Compared to omnivores, shredder-detritivores may exert stronger effects on turnover and decomposition of organic material within lotic systems

The Contractible Subgraph of 5-Connected Graphs and Its Minors

图的连通性是图论的一个重要的研究课题，它对图论的发展有着重大的影响和推动作用.随着大规模计算机网络和通信网络技术的迅速发展，图的连通性的研究与网络可靠性和网络优化的联系日益密切，使得它有了直接的应用背景和应用价值. 可收缩边是研究连通图的构造的强有力工具，在使用归纳法证明图的性质时也起着重要的作用.人们已经知道阶至少为5的3-连通图一定存在3-可收缩边.对于，存在无限多不含可收缩边的k-连通图，即收缩临界k-连通图.由于收缩临界图的任意一条边都不可收缩，于是人们将这个概念推广，定义了k-可收缩子图.M.Kriesell有如下猜想: Conjecture1对于正整数k，存在b(k)和...The connectivity of graph is a key topic in the research of graph theory, as it plays a significantly important role in the development of graph theory. With the development of large scale computer and communication network, it is getting more and more close connection to the reliability of network and optimization of network. This make the connectivity of graph has a directly practical applica...学位：理学博士院系专业：数学科学学院数学与应用数学系_应用数学学号：1902007015385

Edges not contained in triangles and the distribution of contractible edges in a 4-connected graph

AbstractWe prove results concerning the distribution of 4-contractible edges in a 4-connected graph G in connection with the edges of G not contained in a triangle. As a corollary, we show that if G is 4-regular 4-connected graph, then the number of 4-contractible edges of G is at least one half of the number of edges of G not contained in a triangle