 NSFPAR ID:
 10351571
 Date Published:
 Journal Name:
 Young Researcher's Forum (CG Week)
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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The Frechet distance is often used to measure distances between paths, with applications in areas ranging from map matching to GPS trajectory analysis to hand writing recognition. More recently, the Frechet distance has been generalized to a distance between two copies of the same graph embedded or immersed in a metric space; this more general setting opens up a wide range of more complex applications in graph analysis. In this paper, we initiate a study of some of the fundamental topological properties of spaces of paths and of graphs mapped to R^n under the Frechet distance, in an effort to lay the theoretical groundwork for understanding how these distances can be used in practice. In particular, we prove whether or not these spaces, and the metric balls therein, are pathconnected.more » « less

The Fréchet distance is often used to measure distances between paths, with applications in areas ranging from map matching to GPS trajectory analysis to hand writing recognition. More recently, the Fréchet distance has been generalized to a distance between two copies of the same graph embedded or immersed in a metric space; this more general setting opens up a wide range of more complex applications in graph analysis. In this paper, we initiate a study of some of the fundamental topological properties of spaces of paths and of graphs mapped to R^n under the Fréchet distance, in an eort to lay the theoretical groundwork for understanding how these distances can be used in practice. In particular, we prove whether or not these spaces, and the metric balls therein, are pathconnected.more » « less

In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a single path; we call this the trivial space of directed paths. Directed spaces that are topologically trivial may have nontrivial spaces of directed paths, which means that information is lost when the direction of these topological spaces is ignored. We define a notion of directed collapsibility in the setting of a directed Euclidean cubical complex using the spaces of directed paths of the underlying directed topological space, relative to an initial or a final vertex. In addition, we give sufficient conditions for a directed Euclidean cubical complex to have a contractible or a connected space of directed paths from a fixed initial vertex. We also give sufficient conditions for the path space between two vertices in a Euclidean cubical complex to be disconnected. Our results have applications to speeding up the verification process of concurrent programming and to understanding partial executions in concurrent programs.more » « less

Abstract A triangle‐path in a graph is a sequence of distinct triangles in such that for any with , and if . A connected graph is triangularly connected if for any two nonparallel edges and there is a triangle‐path such that and . For ordinary graphs, Fan et al. characterize all triangularly connected graphs that admit nowhere‐zero 3‐flows or 4‐flows. Corollaries of this result include the integer flow of some families of ordinary graphs, such as locally connected graphs due to Lai and some types of products of graphs due to Imrich et al. In this paper, Fan's result for triangularly connected graphs is further extended to signed graphs. We proved that a flow‐admissible triangularly connected signed graph admits a nowhere‐zero 4‐flow if and only if it is not the wheel associated with a specific signature. Moreover, this result is sharp since there are infinitely many unbalanced triangularly connected signed graphs admitting a nowhere‐zero 4‐flow but no 3‐flow.

For graphs G and H, we say that G is Hfree if it does not contain H as an induced subgraph. Already in the early 1980s Alekseev observed that if H is connected, then the Max Weight Independent Set problem (MWIS) remains NPhard in Hfree graphs, unless H is a path or a subdivided claw, i.e., a graph obtained from the threeleaf star by subdividing each edge some number of times (possibly zero). Since then determining the complexity of MWIS in these remaining cases is one of the most important problems in algorithmic graph theory. A general belief is that the problem is polynomialtime solvable, which is witnessed by algorithmic results for graphs excluding some small paths or subdivided claws. A more conclusive evidence was given by the recent breakthrough result by Gartland and Lokshtanov [FOCS 2020]: They proved that MWIS can be solved in quasipolynomial time in Hfree graphs, where H is any fixed path. If H is an arbitrary subdivided claw, we know much less: The problem admits a QPTAS and a subexponentialtime algorithm [Chudnovsky et al., SODA 2019]. In this paper we make an important step towards solving the problem by showing that for any subdivided claw H, MWIS is polynomialtime solvable in Hfree graphs of bounded degree.more » « less