Paths Of Fixed Length Research Articles

Harmonious Labelings of Families of Disjoint Unions of an Odd Cycle and Certain Trees

A graph \(G\) with vertex set \(V = V(G)\) and edge set \(E = E(G)\) is harmonious if there exists a harmonious labeling of \(G\); which is an injective function \(f:V(G) \rightarrow \mathbf{Z}_m\) provided that whenever \(e_1, e_2 \in E\) are distinct with endpoints \(u_1,v_1\) and \(u_2,v_2\), respectively, then \(f(u_1) + f(v_1) \not\equiv f(u_2) + f(v_2) (\hbox{mod } m )\). Using basic group theory, we prove in a different manner an already established result that a disjoint union of an odd cycle and a path is harmonious provided their lengths satisfy certain conditions. We apply the same basic idea to establish that, under the same conditions, a disjoint union of an odd cycle with a certain starlike tree is harmonious (where a starlike tree consists of a central vertex that is adjacent to an endpoint of a certain number of fixed length paths). Finally, we extend the latter result to include specifying that the central vertex in the tree be adjacent to different vertices in each of the \(t\)-many \(s\)-paths.

Game theoretic centrality of a directed graph vertices

The paper considers a game theory approach to calculating the centrality value of the vertices in a directed graph, based on the number of vertex occurrences in fixed length paths. It is proposed to define vertex centrality as a solution of a cooperative game, where the characteristic function is given as the number of simple paths of fixed length in subgraphs corresponding to coalitions. The concept of integral centrality is introduced as the value of a definite integral of the payoff function. It is shown that this centrality measure satisfies the Boldi-Vigna axioms.

Game-Theoretic Centrality of Directed Graph Vertices

The paper considers a game theory approach to calculating the centrality value of the vertices in a directed graph, based on the number of vertex occurrences in fixed length paths. It is proposed to define vertex centrality as a solution of a cooperative game, where the characteristic function is given as the number of simple paths of fixed length in subgraphs corresponding to coalitions. The concept of integral centrality is introduced as the value of a definite integral of the payoff function. It is shown that this centrality measure satisfies the Boldi-Vigna axioms.

Automatic graph-based method for localization of cochlear implant electrode arrays in clinical CT with sub-voxel accuracy.

Cochlear implants (CIs) are neural prosthetics that provide a sense of sound to people who experience severe to profound hearing loss. Recent studies have demonstrated a correlation between hearing outcomes and intra-cochlear locations of CI electrodes. Our group has been conducting investigations on this correlation and has been developing an image-guided cochlear implant programming (IGCIP) system to program CI devices to improve hearing outcomes. One crucial step that has not been automated in IGCIP is the localization of CI electrodes in clinical CTs. Existing methods for CI electrode localization do not generalize well on large-scale datasets of clinical CTs implanted with different brands of CI arrays. In this paper, we propose a novel method for localizing different brands of CI electrodes in clinical CTs. We firstly generate the candidate electrode positions at sub-voxel resolution in a whole head CT by thresholding an up-sampled feature image and voxel-thinning the result. Then, we use a graph-based path-finding algorithm to find a fixed-length path that consists of a subset of the candidates as the localization result. Validation on a large-scale dataset of clinical CTs shows that our proposed method outperforms the state-of-art CI electrode localization methods and achieves a mean error of 0.12 mm when compared to expert manual localization results. This represents a crucial step in translating IGCIP from the laboratory to large-scale clinical use.

The weight of time: Gravitational force enhances discrimination of visual motion duration

In contrast with the anisotropies in spatial and motion vision, anisotropies in the perception of motion duration have not been investigated to our knowledge. Here, we addressed this issue by asking observers to judge the duration of motion of a target accelerating over a fixed length path in one of different directions. Observers watched either a pictorial or a quasi-blank scene, while being upright or tilted by 45° relative to the monitor and Earth's gravity. Finally, observers were upright and we tilted the scene by 45°. We found systematic anisotropies in the precision of the responses, the performance being better for downward motion than for upward motion relative to the scene both when the observer and the scene were upright and when either the observer or the scene were tilted by 45°, although tilting decreased the size of the effect. We argue that implicit knowledge about gravity force is incorporated in the neural mechanisms computing elapsed time. Furthermore, the results suggest that the effects of a virtual gravity can be represented with respect to a vertical direction concordant with the visual scene orientation and discordant with the direction of Earth's gravity.

Quantum snake walk on graphs

I introduce a continuous-time quantum walk on graphs called the quantum snake walk, the basis states of which are fixed-length paths (snakes) in the underlying graph. First, I analyze the quantum snake walk on the line, and I show that, even though most states stay localized throughout the evolution, there are specific states that most likely move on the line as wave packets with momentum inversely proportional to the length of the snake. Next, I discuss how an algorithm based on the quantum snake walk might potentially be able to solve an extended version of the glued trees problem, which asks to find a path connecting both roots of the glued trees graph. To the best of my knowledge, no efficient quantum algorithm solving this problem is known yet.

Improving a nonenumerative method to estimate path delay fault coverage

A recently proposed method obtains path delay fault coverages by estimating the count of the number of tested faults instead of actually enumerating them. The estimate becomes pessimistic when several paths share a set of lines. In this communication, we present a continuum of improved approximations for the counting method, approaching exact fault simulation, to allow tradeoffs between accuracy and complexity. Higher accuracy is obtained at the expense of CPU time. We propose the use of flags corresponding to fixed-length path segments. A flag indicates whether or not the segment has been included in a previously detected path fault. A path fault, detected by a pair of vectors, is counted as a new detection only if it includes at least one segment not included in any previously tested path fault. The results show that as the length of segments is increased, the accuracy becomes close to that of the exact fault simulation, even with small segment lengths.