A subset S of vertices of a graph G = (V,E) is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from S. Denote by ψk(G) the minimum cardinality of a k-path vertex cover in G and form a sequence ψ(G) = (ψ1(G), ψ2(G), . . . , ψ|V |(G)), called the path sequence of G. In this paper we prove necessary and sufficient conditions for two integers to appear on ...

Personalized learning paths are becoming more common in adaptive learning systems. In this paper we extend our earlier work by developing the Content Path Combination Model (CPCM) for integrating the design paths obtained from different instructional designers and making a strongly recommended course object sequence; presenting methodologies for handling two central tasks: (i) creating the line...

We give a minimal change list for the set of order p length-n Lucas strings, i.e., the set of length-n binary stringswith no p consecutive 1’s nor a 1 prefix and a 1m suffixwith +m p. The construction of this list proves also that the order p n-dimensional Lucas cube has a Hamiltonian path if and only if n is not a multiple of p + 1, and its second power always has a Hamiltonian path. © 2005 El...

This paper presents a mission planning method for planetary rover. As a decision support system, mission planning is important for rover navigation and control. Generating travel path and task sequence synchronously is the key point. In this paper, the environment model of planetary surface is summed up as two parts which are obstacle map and timeline. According to this, the mission planner uti...

The paper proposes algorithms for collision-free path planning in robotic Remote Laser Welding (RLW), using collision detection on a triangle mesh representation of the moving objects and a path planning algorithm based on a classical A∗ search, both highly specialized to the needs of RLW. The algorithms depart from an optimized task sequence and an initial, potentially colliding rough-cut path...

A tree is a structure T = (T,E), where T is a nonempty set whose elements are called nodes and E is a relation on T , E ⊆ T × T , called the immediate edge relation, satisfying the following conditions: for all nodes, n, n′,m ∈ T , • Every node has a unique predecessor: If nEm and n′Em, then n = n′, • There are no cycles: If (n1, . . . , nk) is a sequence of nodes where for each i = 1, . . . , ...

Let T be a rooted tree. A path reversal at a node x in T is performed by traversing the path from x to the tree root r and making x the parent of each node on the path other than X. Thus x becomes the new tree root. (See Fig. 1). The cost of the reversal is the number of edges on the path reversed. Path reversal is a variant of the standard path compression algorithm for maintaining disjoint se...

The present study investigates the pendant number of certain graph classes; complement, line graphs, and total graphs. is minimum end vertices paths in a path decomposition graph. A it into subgraphs; i.e., sequence subset such that endpoints each edge appear one subsets vertex appears an adjacent sub-sequence subsets.