We consider the problem of maintaining a large matching and a small vertex cover in a dynamically changing graph. Each update to the graph is either an edge deletion or an edge insertion. We give the first randomized data structure that simultaneously achieves a constant approximation factor and handles a sequence of K updates in K · polylog(n) time, where n is the number of vertices in the gra...

Proof. The first equality follows directly from (1). To see the second equality, first let M be a matching of size ν(G). For each of the |V | − 2|M | vertices v missed by M , add to M an edge covering v. We obtain an edge cover of size |M |+ (|V | − 2|M |) = |V | − |M |. Hence ρ(G) ≤ |V | − ν(G). Second, let F be an edge cover of size ρ(G). For each v ∈ V delete from F , dF (v) − 1 edges incide...

Every 2-connected simple cubic graph of order n ≥ 6 has a cycle cover with at most dn4 e cycles. This bound is best possible.

A cost-sharing mechanism defines how to share the cost of a service among serviced customers. It solicits bids from potential customers and selects a subset of customers to serve and a price to charge each of them. The mechanism is group-strategyproof if no subset of customers can gain by lying about their values. There is a rich literature that designs group-strategyproof cost-sharing mechanis...

A graph is called pseudo-outerplanar if each block has an embedding on the plane in such a way that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, we prove that each pseudo-outerplanar graph admits edge decompositions into a linear forest and an outerplanar graph, or a star forest and an outerpla...

The reload cost refers to the cost that occurs along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. Galbiati et al. [1] introduced the Minimum Reload Cost Cycle Cover problem, which is to find a set of vertex-disjoint cycles spanning all vertices with minimum reload cost. They proved that this problem is strongly NP-hard and not appro...

We study the approximability of the weighted edge-dominating set problem. Although even the unweighted case is NP-Complete, in this case a solution of size at most twice the minimum can be efficiently computed due to its close relationship with minimum maximal matching; however, in the weighted case such a nice relationship is not known to exist. In this paper, after showing that weighted edge ...

We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and deletion of edges together with their end points. Like in the case of deletion and contraction only (W. Tutte, 1954), it turns out that there is a most general polynomial satisfying such recurrence relations, which we call ξ(G, x, y, z). We show that the new p...

The paper is about the edge covers of the graph. The change in the edge covering number of the graph is observed under the operations vertex removal, edge removal and edge addition. The necessary and sufficient condition is proved under which the edge covering number decreases when a vertex is removed from the graph also when an edge is added to the graph. The necessary and sufficient condition...