A graph theory-based multi-scale analysis of hierarchical cascade in molecular clouds : Application to the NGC 2264 region

The spatial properties of small star-clusters suggest that they may originate from a fragmentation cascade of the cloud for which there might be traces up to a few dozen of kAU. Our goal is to investigate the multi-scale spatial structure of gas clumps, to probe the existence of a hierarchical cascade and to evaluate its possible link with star production in terms of multiplicity. From the Herschel emission maps of NGC 2264, clumps are extracted using getsf software at each of their associated spatial resolution, respectively [8.4, 13.5, 18.2, 24.9, 36.3]". Using the spatial distribution of these clumps and the class 0/I Young Stellar Object (YSO) from Spitzer data, we develop a graph-theoretic analysis to represent the multi-scale structure of the cloud as a connected network. From this network, we derive three classes of multi-scale structure in NGC 2264 depending on the number of nodes produced at the deepest level: hierarchical, linear and isolated. The structure class is strongly correlated with the column density $N_{\rm H_2}$ since the hierarchical ones dominate the regions whose N$_{\rm H_2} > 6 \times 10^{22}$cm$^{-2}$. Although the latter are in minority, they contain half of the class 0/I YSOs proving that they are highly efficient in producing stars. We define a novel statistical metric, the fractality coefficient F that measure the fractal index describing the scale-free process of the cascade. For NGC 2264, we estimate F = 1.45$\pm$0.12. However, a single fractal index fails to fully describe a scale-free process since the hierarchical cascade starts at a 13 kAU characteristic spatial scale. Our novel methodology allows us to correlate YSOs with their multi-scale gaseous environment. This hierarchical cascade that drives efficient star formation is suspected to be both hierarchical and rooted by the larger-scale gas environment up to 13 kAU

Fully dynamic recognition of proper circular-arc graphs

We present a fully dynamic algorithm for the recognition of proper circular-arc (PCA) graphs. The allowed operations on the graph involve the insertion and removal of vertices (together with its incident edges) or edges. Edge operations cost O(log n) time, where n is the number of vertices of the graph, while vertex operations cost O(log n + d) time, where d is the degree of the modified vertex. We also show incremental and decremental algorithms that work in O(1) time per inserted or removed edge. As part of our algorithm, fully dynamic connectivity and co-connectivity algorithms that work in O(log n) time per operation are obtained. Also, an O(\Delta) time algorithm for determining if a PCA representation corresponds to a co-bipartite graph is provided, where \Delta\ is the maximum among the degrees of the vertices. When the graph is co-bipartite, a co-bipartition of each of its co-components is obtained within the same amount of time.Comment: 60 pages, 15 figure

Partial Homology Relations - Satisfiability in terms of Di-Cographs

Directed cographs (di-cographs) play a crucial role in the reconstruction of evolutionary histories of genes based on homology relations which are binary relations between genes. A variety of methods based on pairwise sequence comparisons can be used to infer such homology relations (e.g.\ orthology, paralogy, xenology). They are \emph{satisfiable} if the relations can be explained by an event-labeled gene tree, i.e., they can simultaneously co-exist in an evolutionary history of the underlying genes. Every gene tree is equivalently interpreted as a so-called cotree that entirely encodes the structure of a di-cograph. Thus, satisfiable homology relations must necessarily form a di-cograph. The inferred homology relations might not cover each pair of genes and thus, provide only partial knowledge on the full set of homology relations. Moreover, for particular pairs of genes, it might be known with a high degree of certainty that they are not orthologs (resp.\ paralogs, xenologs) which yields forbidden pairs of genes. Motivated by this observation, we characterize (partial) satisfiable homology relations with or without forbidden gene pairs, provide a quadratic-time algorithm for their recognition and for the computation of a cotree that explains the given relations

A fully dynamic algorithm for the recognition of P4-sparse graphs

In this paper, we solve the dynamic recognition problem for the class of P4sparse graphs: the objective is to handle edge/vertex additions and deletions, to recognize if each such modification yields a P4-sparse graph, and if yes, to update a representation of the graph. Our approach relies on maintaining the modular decomposition tree of the graph, which we use for solving the recognition problem. We establish properties for each modification to yield a P4-sparse graph and obtain a fully dynamic recognition algorithm which handles edge modifications in O(1) time and vertex modifications in O(d) time for a vertex of degree d. Thus, our algorithm implies an optimal edges-only dynamic algorithm and a new optimal incremental algorithm for P4-sparse graphs