We observe that ω(G) + χ(S( G)) = n = ω(S( G)) + χ(G) for any graph G with n vertices, where G is any acyclic orientation of G and where S( G) is the (complement of the) auxiliary line graph introduced in [1]. (Where as usual, ω and χ denote the clique number and the chromatic number.) It follows that, for any graph parameter β(G) sandwiched between ω(G) and χ(G), then Φβ( G) := n−β(S( G)) is s...

Recent applications in computer vision have come to rely on superpixel segmentation as a pre-processing step for higher level vision tasks, such as object recognition, scene labelling or image segmentation. Here, we present a new algorithm, Superpixels from MUlti-scale ReFinement of Super-regions (SMURFS), which not only obtains state-ofthe-art superpixels, but can also be applied hierarchicall...

Graphs have been widely used to model different information networks, such as the Web, biological networks and social networks (e.g. Twitter). Due to the size and complexity of these graphs, how to explore and utilize these graphs has become a very challenging problem. In this paper, we propose, VCExplorer, a new interactive graph exploration framework that integrates the strengths of graph vis...

Encoding the clustering result to represent the structure feature of a tree ‹ Encoding each cluster as following field ‹Link all cluster fields to make a long numeric array that can represent the structure of a tree Describing the trees' super‐sub relationships with a network, and using a numeric method to access it fast. PURPOSE ‡ Structure Network: A network describing the super‐sub relations...

The paper presents a design for the Denali-2 super-optimizer, which will generate minimum-instruction-length machine code for realistic machine architectures using automatic theorem-proving technology: specifically, using E-graph matching (a technique for pattern matching in the presence of equality information) and boolean satisfiability solving. The paper presents a precise definition of the ...

In side-scrolling games such as Super Mario Bros, the player interacts with a simplified, two-dimensional world made up of elements from a finite set. Conventionally, specialized designers build the levels of the game, applying their creativity and experience to produce content exhibiting structural correctness (the player must be able to traverse the level from start to end), interestingness a...

Let G be a finite graph with p vertices and q edges. A vertex magic total labelling is a bijection from the union of the vertex set and the edge set to the consecutive integers 1, 2, 3, . . . , p + q with the property that for every u in the vertex set, the sum of the label of u and the label of the edges incident with u is equal to k for some constant k. Such a labelling is E-super, if the lab...

Let G be a connected graph with minimum degree δ and edgeconnectivity λ. A graph is maximally edge-connected if λ = δ, and it is superedge-connected if every minimum edge-cut is trivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree. In this paper, we show that a connected graph or a connected triangle-free graph is maximally edge-connected or sup...

An edge magic total labeling of a graph G(V,E) with p vertices and q edges is a bijection f from the set of vertices and edges to such that for every edge uv in E, f(u) + f(uv) + f(v) is a constant k. If there exist two constants k1 and k2 such that the above sum is either k1 or k2, it is said to be an edge bimagic total labeling. A total edge magic (edge bimagic) graph is called a super edge m...

Restricted edge connectivity is a more refined network reliability index than edge connectivity. A restricted edge cut F of a connected graph G is an edge cut such that G−F has no isolated vertex. The restricted edge connectivity λ′ is theminimumcardinality over all restricted edge cuts.WecallG λ′-optimal if λ′ = ξ , where ξ is theminimum edgedegree inG. Moreover, a λ′-optimal graphG is called ...