On the hamiltonian path graph of a graph

The hamiltonian path graph H(F) of a graph F is that graph having the same vertex set as F and in which two vertices u and v are adjacent if and only if F contains a hamiltonian u ? v path. First, in response to a conjecture of Chartrand, Kapoor and Nordhaus, a characterization of nonhamiltonian graphs isomorphic to their hamiltonian path graphs is presented. Next, the maximum size of a hamiltonian graph F of given order such that K?_d ? H(F) is determined. Finally, it is shown that if the degree sum of the endvertices of a hamiltonian path in a graph F with at least five vertices is at least |V(F)| + t(t ? 0), then H(F) contains a complete subgraph of order t + 4.     

Regular graphs having the same path layer matrix

The path layer matrix (or path degree sequence) of a graph G contains quantitative information about all paths in G. Elements (i,j) in this matrix is the number of simple paths in G having initial vertex v, and length j. For every r ≥ 3, pairs of nonisomorphic r-regular graphs having the same path layer matrix are presented.     

On Partitioning the Edge Set of a Graph into Internally Disjoint Paths without Exterior Vertices

A collection of nontrivial paths in a graph G is called a path pile of G, if every edge of G is on exactly one path and no two paths have a common internal vertex. The least number that can be the cardinality of a path pile of G is called the path piling number of G. It can be shown that ε ≤ ν + η where ε, ν and η are respectively the size, the order and the path piling number of G. In this note we characterize structurally the class of all graphs for which the equality of this relation holds.     

On the K shortest path trees problem

We address the problem of finding the K best path trees connecting a source node with any other non-source node in a directed network with arbitrary lengths. The main result in this paper is the proof that the kth shortest path tree is adjacent to at least one of the previous (k-1) shortest path trees. Consequently, we design an O(f(n,m,C_max)+Km) time and O(K+m) space algorithm to determine the K shortest path trees, in a directed network with n nodes, m arcs and maximum absolute length C_max, where O(f(n,m,C_max)) is the best time needed to solve the shortest simple paths connecting a source node with any other non-source node.     

Continuity of the straight line path for convex sets

IfA andB are closed nonempty sets in a locally convex space, the straight line path fromA toB is defined by the formulaφ(α)=cl (αA+(1−α)B), 0≦α≦1. IfA andB are convex, then continuity of the path with respect to the Hausdorff uniform topology is necessary for both connectedness and path connectedness ofA toB within the convex sets so topologized. We also produce internal necessary and sufficient conditions for continuity of the path between pairs of convex sets.     

Cubic graphs with 62 vertices having the same path layer matrix

The path layer matrix (or path degree sequence) of a graph G contains quantitative information about all paths in G. The entry (i,j) in this matrix is the number of simple paths in G having initial vertex v and length j. It was known that there are cubic graphs on 90 vertices having the same path layer matrix (Dobrynin, 1990). A new upper bound for the least order of cubic graphs with the same path layer matrix is presented. © 1993 John Wiley & Sons, Inc.     

Rank and biclique partitions of the complement of paths

The problem was posed of determining the biclique partition number of the complement of a Hamiltonian path (Monson, Rees, and Pullman, Bull. Inst. Combinatorics and Appl. 14 (1995), 17–86). We define the complement of a path P, denoted P , as the complement of P in K_m,n where P is a subgraph of K_m,n for some m and n. We give an exact formula for the biclique partition number of the complement of a path. In particular, we solve the problem posed in [9]. We also summarize our more general results on biclique partitions of the complement of forests. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 111–122, 1998     

Paths extendable to cycles

Let k be a positive integer, and S a nonempty set of positive integers. Suppose that G is a connected graph containing a path of length k, and that each path P of length k in G is contained in some cycle C(P) of length s ∈ S. We prove that every path of length less than k can be extended to a path of length k in G. This result answers conjectures of Entringer and Reid regarding when certain paths may be extended to cycles.     

Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groups

In this paper the concepts of Hamilton cycle (HC) and Hamilton path (HP) extendability are introduced. A connected graph Γ is n‐HC‐extendable if it contains a path of length n and if every such path is contained in some Hamilton cycle of Γ. Similarly, Γ is weakly n‐HP‐extendable if it contains a path of length n and if every such path is contained in some Hamilton path of Γ. Moreover, Γ is strongly n‐HP‐extendable if it contains a path of length n and if for every such path P there is a Hamilton path of Γ starting with P. These concepts are then studied for the class of connected Cayley graphs on abelian groups. It is proved that every connected Cayley graph on an abelian group of order at least three is 2‐HC‐extendable and a complete classification of 3‐HC‐extendable connected Cayley graphs of abelian groups is obtained. Moreover, it is proved that every connected Cayley graph on an abelian group of order at least five is weakly 4‐HP‐extendable. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory     

Rectilinear shortest paths in the presence of rectangular barriers

In this paper we address the following shortest-path problem. Given a point in the plane andn disjoint isothetic rectangles (barriers), we want to construct a shortestL ₁ path (not crossing any of the barriers) from the source point to any given query point. A restricted version of this problem (where the source and destination points are knowna priori) had been solved earlier inO(n ²) time. Our approach consists of preprocessing the source point and the barriers to obtain a planar subdivision where a query point can be located and a shortest path connecting it to the source point quickly transvered. By showing that any such path is monotone in at least one ofx ory directions, we are able to apply a plane sweep technique to divide the plane intoO(n) rectangular regions. This leads to an algorithm whose complexity isO(n logn) preprocessing time,O(n) space, andO(logn+k) query time, wherek is the number of turns on the reported path. If only the length of the path is sought,O(logn) query time suffices. Furthermore, we show an (n logn) time lower bound for the case where the source and destination points are known in advance, which implies the optimality of our algorithm in this case.A preliminary version of this paper appeared in theProceedings of the First Symposium on Computational Geometry (1985).Supported in part by CNPq-Conselho Nacional de Desenvolvimento Científico e Tecnológico (Brazil).Supported in part by the National Science Foundation under Grants MCS 8420814 and ECS 8340031.     

Upper Bounds for the Rainbow Connection Numbers of Line Graphs

A path in an edge-colored graph G, where adjacent edges may be colored the same, is called a rainbow path if no two edges of it are colored the same. A nontrivial connected graph G is rainbow connected if for any two vertices of G there is a rainbow path connecting them. The rainbow connection number of G, denoted rc(G), is defined as the smallest number of colors such that G is rainbow connected. In this paper, we mainly study the rainbow connection number rc(L(G)) of the line graph L(G) of a graph G which contains triangles. We get two sharp upper bounds for rc(L(G)), in terms of the number of edge-disjoint triangles of G. We also give results on the iterated line graphs.     

Complexity analysis and optimization of the shortest path tour problem

The shortest path tour problem (SPTP) consists in finding a shortest path from a given origination node s to a given destination node d in a directed graph with nonnegative arc lengths with the constraint that the optimal path P should successively and sequentially pass through at least one node from given node subsets T ₁, T ₂, . . . , T _N , where T_i ?T_j = ?, " i, j=1,?,N, i 1 j{T_i \cap T_j = \emptyset, \forall\ i, j=1,\ldots,N,\ i \neq j}. In this paper, it will proved that the SPTP belongs to the complexity class P and several alternative techniques will be presented to solve it.     

The spanning trees forced by the path and the star

A set S of trees of order n forces a tree T if every graph having each tree in S as a spanning tree must also have T as a spanning tree. A spanning tree forcing set for order n that forces every tree of order n. A spanning-tree forcing set S is a test set for panarboreal graphs, since a graph of order n is panarboreal if and only if it has all of the trees in S as spanning trees. For each positive integer n ≠ 1, the star belongs to every spanning tree forcing set for order n. The main results of this paper are a proof that the path belongs to every spanning-tree forcing set for each order n ∉ {1, 6, 7, 8} and a computationally tractable characterization of the trees of order n ≥ 15 forced by the path and the star. Corollaries of those results include a construction of many trees that do not belong to any minimal spanning tree forcing set for orders n ≥ 15 and a proof that the following related decision problem is NP-complete: an instance is a pair (G, T) consisting of a graph G of order n and maximum degree n - 1 with a hamiltonian path, and a tree T of order n; the problem is to determine whether T is a spanning tree of G. © 1996 John Wiley & Sons, Inc.     

Hamiltonian path graphs

The Hamiltonian path graph H(G) of a graph G is that graph having the same vertex set as G and in which two vertices u and v are adjacent if and only if G contains a Hamiltonian u-v path. A characterization of Hamiltonian graphs isomorphic to their Hamiltonian path graphs is presented.     

Path decompositions of multigraphs

Let G be a loopless finite multigraph. For each vertex x of G, denote its degree and multiplicity by d(x) and μ(x) respectively. Define Ø(x) = the least even integer ≥ μ(x), if d(x) is even, the least odd integer ≥ μ(x), if d(x) is odd. In this paper it is shown that every multigraph G admits a faithful path decomposition—a partition P of the edges of G into simple paths such that every vertex x of G is an end of exactly Ø(x) paths in P. This result generalizes Lovász's path decomposition theorem, Li's perfect path double cover theorem (conjectured by Bondy), and a result of Fan concerning path covers of weighted graphs. It also implies an upper bound on the number of paths in a minimum path decomposition of a multigraph, which motivates a generalization of Gallai's path decomposition conjecture. © 1995 John Wiley & Sons, Inc.     

On finding the core of a tree with a specified length

A core of a graph G is a path P in G that is central with respect to the property of minizining d(P) = Σ_υεV(G)d(υ, P), where d(υ, P) is the distance from vertex υ to path P. This paper explores some properties of a core of a specified length.     

Planar rectilinear shortest path computation using corridors

The rectilinear shortest path problem can be stated as follows: given a set of m non-intersecting simple polygonal obstacles in the plane, find a shortest L₁-metric (rectilinear) path from a point s to a point t that avoids all the obstacles. The path can touch an obstacle but does not cross it. This paper presents an algorithm with time complexity O(n+m(lgn)^3/2), which is close to the known lower bound of Ω(n+mlgm) for finding such a path. Here, n is the number of vertices of all the obstacles together.     

*-Regular Leavitt path algebras of arbitrary graphs

If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra L _K (E). We show that the involution on L _K (E) is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra L _K (E) is regular if and only if E is acyclic. We give necessary and sufficient conditions for L _K (E) to be *-regular (i.e., regular with proper involution). This characterization of *-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graphtheoretic properties of E alone. As a corollary, we show that Handelman’s conjecture (stating that every *-regular ring is unit-regular) holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.     

Cubic graphs without cut‐vertices having the same path layer matrix

The path layer matrix (or path degree sequence) of a graph G contains quantitative information about all possible paths in G. The entry (i,j) of this matrix is the number of paths in G having initial vertex i and length j. It is known that there are cubic graphs on 62 vertices having the same path layer matrix (A. A. Dobrynin. J Graph Theory 17 (1993) 1–4). A new upper bound of 36 vertices for the least order of such cubic graphs is established. This bound is realized by cubic graphs without cut‐vertices. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 177–182, 2001     

Optimal algorithm for the nearest common dominator problem

In a directed acyclic graph G = (V, E) with a single source s, the nearest common dominator for a set of nodes U V is the node d closest to U such that every path from s to any node in U goes through d. The distance between any node w and U is defined as the number of arcs in a shortest path from w to any node in U. We develop an optimal algorithm for finding the nearest common dominator d for U, which runs in time proportional to the number of arcs in between d and U.