We study the maximum flow problem in an undirected planar network with both edge and vertex capacities (EVC-network). A previous study reduces the minimum cut problem in an undirected planar EVC-network to the minimum edge-cut problem in another planar network with edge capacity only (EC-network), thus the minimum-cut or the maximum flow value can be computed in O(n log n) time. Based on this r...

Let C be a uniform clutter and let I = I(C) be its edge ideal. We prove that if C satisfies the packing property (resp. max-flow min-cut property), then there is a uniform Cohen-Macaulay clutter C1 satisfying the packing property (resp. max-flow min-cut property) such that C is a minor of C1. For arbitrary edge ideals of clutters we prove that the normality property is closed under parallelizat...

The aim of this study was to determine why food poisoning bacteria attached to cut cabbage are not efficiently disinfected by sodium hypochlorite (NaClO). Pretreatment of shredded cabbage with diethyl ether definitely decreased the survival numbers of Escherichia coli O157:H7 and Salmonella spp. after disinfection with 100 ppm of NaClO. The density of E. coli O157:H7 at the cut edge of a cabbag...

For a connected graphG= (V ,E), an edge set S ⊂ E is a k-restricted-edge-cut, ifG−S is disconnected and every component of G− S has at least k vertices. The k-restricted-edge-connectivity of G, denoted by k(G), is defined as the cardinality of a minimum k-restricted-edge-cut. The k-isoperimetric-edge-connectivity is defined as k(G)=min{|[U,U ]| : U ⊂ V, |U | k, |U | k}, where [U,U ] is the set ...

Many polynomially solvable combinatorial optimization problems (COP) become NP hard when we require solutions to satisfy an additional cardinality constraint. This family of problems has been considered only recently. We study a new problem of this family: the k-cardinality minimum cut problem. Given an undirected edge-weighted graph the k-cardinality minimum cut problem is to nd a partition of...

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 c...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

In this note we report on our recent work, still in progress, regarding Folkman numbers. Let f(2, 3, 4) denote the smallest integer n such that there exists a K4– free graph of order n having that property that any 2–coloring of its edges yields at least one monochromatic triangle. It is well–known that such a number must exist [4,10]. For almost twenty years the best known upper bound, given b...