A biased graph consists of a graph G together with a collection of distinguished cycles of G, called balanced, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs on G arise from orienting G and then labelling the edges of G with elements of a group Γ. In this case, we may define a biased graph by declaring a cycle to be balanced...

We present python libraries for Feynman graphs manipulation. The key feature of these libraries is usage of generalization of graph representation offered by B. G. Nickel et al. In this approach graph is represented in some unique ’canonical’ form that depends only on its combinatorial type. The uniqueness of graph representation gives an efficient way for isomorphism finding, searching for sub...

In this paper we prove that arbitrary Supersubdivision of disconnected graph is graceful.

Image semantic segmentation is the task of associating a semantic category label to each image pixel. This classification problem is characterised by pixel dependencies at different scales. On a small-scale pixel correlation is related to object instance sharing, while on a middleand large-scale to category co-presence and relative location constraints. The contribution of this paper is twofold...

A graceful labelling of a graph with n edges is a vertex labelling where the induced set of edge weights is {1, . . . , n}. A near graceful labelling is almost the same, the difference being that the edge weights are {1, 2, . . . , n − 1, n + 1}. In both cases, the weight of an edge is the absolute difference between its two vertex labels. Rosa [8] in 1988 conjectured that all triangular cacti ...

We show that to each graceful labelling of a path on 2s + 1 vertices, s ≥ 2, there corresponds a current assignment on a 3-valent graph which generates at least 22s cyclic oriented triangular embeddings of a complete graph on 12s + 7 vertices. We also show that in this correspondence, two distinct graceful labellings never give isomorphic oriented embeddings. Since the number of graceful labell...

The Kneser graph Kn:k for positive integers n ≥ k has as its vertex set the k-element subsets of some n-set, with disjoint sets being adjacent. Every finite simple graph can be found as an induced subgraph of some Kneser graph; this can be viewed as a way of representing graphs by labelling their vertices with sets. We explore questions of finding the smallest representation (both in terms of n...

Deterministic graph grammars generate regular graphs, that form a structural extension of configuration graphs of pushdown systems. In this paper, we study a probabilistic extension of regular graphs obtained by labelling the terminal arcs of the graph grammars by probabilities. Stochastic properties of these graphs are expressed using PCTL, a probabilistic extension of computation tree logic. ...

A subgraph $H$ of a graph $G$ is isometric if the distances between vertices in coincide with corresponding $G$. We show that for any integer $n\ge 1$, there on $3^{n+O(\log^2 n)}$ contains copies all $n$-vertex graphs. Our main tool new type distance labelling scheme, whose study might be independent interest.

We consider the following map labelling problem: given distinct points p,p, . . . ,pn in the plane, and given σ, find a maximum cardinality set of pairwise disjoint axis-parallel σ× σ squares Q1, Q2, . . . , Qr. This problem reduces to that of finding a maximum cardinality independent set in an associated graph called the conflict graph. We describe several heuristics for the maximum cardinalit...