Let G = (V,A) be a vertex-colored arc-weighted directed acyclic graph (DAG) rooted in some vertex r. The color hierarchy graph H(G) of G is defined as follows: V (H(G)) is the color set C of G, and H(G) has an arc from c to c if G has an arc from a vertex of color c to a vertex of color c. We study the Maximum Colorful Arborescence (MCA) problem, which takes as input a DAG G such that H(G) is a...

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An essential feature of large scale free graphs, such as the Web, protein-to-protein interaction, brain connectivity, and social media graphs, is that they tend to form recursive communities. The latter are densely connected vertex clusters exhibiting quick local information dissemination and processing. Under the fuzzy graph model vertices are fixed while each edge exists with a given probabil...

The fuzzy graph approach is more powerful in cluster analysis than the usual graph – theoretic approach due to its ability to handle the strengths of arcs effectively. The concept of node-strength sequence is introduced and is studied in a complete fuzzy graph. Two new connectivity parameters in fuzzy graphs namely, fuzzy node connectivity ðjÞ and fuzzy arc connectivity ðj0Þ are introduced and ...

Let G = (V,E) be a vertex-colored graph, where C is the set of colors used to color V . The Graph Motif (or GM) problem takes as input G, a multisetM of colors built from C, and asks whether there is a subset S ⊆ V such that (i) G[S] is connected and (ii) the multiset of colors obtained from S equals M . The Colorful Graph Motif (or CGM) problem is the special case of GM in which M is a set, an...

We live in interesting times: we know dark matter exists, and we now know how much. At the same time, we have very little idea what it is, other than that it can’t be any known particle. Cosmology therefore provides the longawaited quantitative evidence for new particle physics. The dark matter problem is arguably our strongest beacon, an unambiguous and fundamental, yet approachable, problem p...