We investigate the odd multiway node (edge) cut problem where the input is a graph with a specified collection of terminal nodes and the goal is to find a smallest subset of non-terminal nodes (edges) to delete so that the terminal nodes do not have an odd length path between them. In an earlier work, Lokshtanov and Ramanujan showed that both odd multiway node cut and odd multiway edge cut are ...

Orthogonal drawings are widely used for graph visualization due to their high clarity and ease of representation. But when it comes to highdegree and massive graphs, even orthogonal drawings have difficulties producing a clean and simple visualization. In this paper we present a technique called Overloaded Orthogonal Drawing that greatly improves the readability by proposing a new vertex placem...

Given an edge-weighted undirected graph and two vertices s and t, the next-to-shortest path problem is to find an st-path whose length is minimum among all st-paths of lengths strictly larger than the shortest path length. The problem is shown to be polynomially solvable if all edge weights are positive, while the complexity status for the nonnegative weight case was open. In this paper we show...

Let D be an acyclic digraph. The competition graph of D is a graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number o...

In this paper, we present a precise characterization of the existence of a convergent transfer subgraph in an edge colored directed acyclic graph. Based on the characterization, linear time algorithms are proposed to decide the existence of a convergent transfer subgraph and, if one exists, to construct such a subgraph, respectively. The convergent transfer subgraph (CTS) problem arises from th...

For two graphs S and T , the constrained Ramsey number f(S, T ) is the minimum n such that every edge coloring of the complete graph on n vertices (with any number of colors) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T . Here, a subgraph is said to be rainbow if all of its edges have different colors. It is an immediate consequence of the Erdős-Rado Canoni...

We give a new family of Lovász Local Lemmas (LLL), with applications. Shearer has given the most general condition under which the LLL holds, but the original condition of Lovász is simpler and more practical. Do we have to make a choice between practical and optimal? In this article we present a continuum of LLLs between the original and Shearer’s conditions. One of these, which we call Clique...

Given a graph G = (V,E), let P be a partition of V . We say that P is dominating if, for each part P of P, the set V \ P is a dominating set in G (equivalently, if every vertex has a neighbour of a different colour from its own). We say that P is acyclic if for any parts P, P ′ of P, the bipartite subgraph G[P, P ′] consisting of the edges between P and P ′ in P contains no cycles. The acyclic ...

In the classical Maximum Acyclic Subgraph problem (MAS), given a directed-edge weighted graph, we are required to find an ordering of the nodes that maximizes the total weight of forward-directed edges. MAS admits a 2-approximation, and this approximation is optimal under the Unique Game Conjecture. In this paper we consider a generalization of MAS, the Restricted Maximum Acyclic Subgraph probl...

Motivated by the problem of designing inference-friendly Bayesian nonparametric models in probabilistic programming languages, we introduce a general class partially exchangeable random arrays which generalizes notion hierarchical exchangeability introduced Austin and Panchenko (2014). We say that our are DAG-exchangeable since their structure is governed collection Directed Acyclic Graphs. Mor...