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Given a graph $G=(V,E)$, capacities $w(e)$ on edges, and a subset of terminals $mathcal{T} subseteq V: |mathcal{T}| = k$, a mimicking network for $(G,mathcal{T})$ is a graph $(H,w)$ that contains copies of $mathcal{T}$ and preserves the value of minimum cuts separating any subset $A, B subseteq mathcal{T}$ of terminals. Mimicking networks of size $2^{2^k}$ are known to exist and can be constructed algorithmically, while the best known lower bound is $2^{Omega(k)}$; therefore, an exponential size is required if one aims at preserving cuts exactly. In this paper, we study mimicking networks that preserve connectivity of the graph exactly up to the value of $c$, where $c$ is a parameter. This notion of mimicking network is sufficient for some applications, as we will elaborate. We first show that a mimicking of size $3^c cdot k$ exists, that is, we can preserve cuts with small capacity using a network of size linear in $k$. Next, we show an algorithm that finds such a mimicking network in time $2^{O(c^2)} operatorname{poly}(m)$.
We study two variants of textsc{Maximum Cut}, which we call textsc{Connected Maximum Cut} and textsc{Maximum Minimal Cut}, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity
Given a capacitated undirected graph $G=(V,E)$ with a set of terminals $K subset V$, a mimicking network is a smaller graph $H=(V_H,E_H)$ that exactly preserves all the minimum cuts between the terminals. Specifically, the vertex set of the sparsifie
We study the problem of Imbalance parameterized by the twin cover of a graph. We show that Imbalance is XP parameterized by twin cover, and FPT when parameterized by the twin cover and the size of the largest clique outside the twin cover. In contras
After the number of vertices, Vertex Cover is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that are not directly related to the Vertex C
Let $mathbb{F}[X]$ be the polynomial ring over the variables $X={x_1,x_2, ldots, x_n}$. An ideal $I=langle p_1(x_1), ldots, p_n(x_n)rangle$ generated by univariate polynomials ${p_i(x_i)}_{i=1}^n$ is a emph{univariate ideal}. We study the ideal membe