A path cover is a decomposition of the edges of a graph into edge-disjoint simple paths. Gallai conjectured that every connected n-vertex graph has a path cover with at most dn/2e paths. We prove Gallai’s conjecture for series-parallel graphs. For the class of planar 3-trees we show how to construct a path cover with at most b5n/8c paths, which is an improvement over the best previously known b...

Given a tree and a set of paths in the tree, the problem of finding a minimum number of paths from the given path set to cover all the vertices in the tree is investigated in the paper. To distinguish from the classical path cover problem, such an optimization problem is referred to as vertex covering by paths. The problem and its edge variant, edge covering by paths, find applications in machi...

A simple path cover of a graph G is a collection ψ of paths in G such that every edge of G is in exactly one path in ψ and any two paths in ψ have at most one vertex in common. More generally, for any integer k ≥ 1, a Smarandache path k-cover of a graph G is a collection ψ of paths in G such that each edge of G is in at least one path of ψ and two paths of ψ have at most k vertices in common. T...

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a proper interval graph G and a particular set T of k vertices, the goal is to compute a path cover of G with minimum cardinality, such that the vertices of T are endpoints of these paths. We propose an optimal algorithm for this problem with ru...

Given a graph G, and two vertex sets S and T of size k each, a many-tomany k-disjoint path cover of G joining S and T is a collection of k disjoint paths between S and T that cover every vertex of G. It is classified as paired if each vertex of S must be joined to a designated vertex of T , or unpaired if there is no such constraint. In this article, we first present a necessary and sufficient ...

A chord of a path $P$ is an edge joining two non-adjacent vertices of $P$. A path  $P$ is called a monophonic path if it is a chordless path. A longest $x-y$ monophonic path is called an $x-y$ detour monophonic path. A  detour monophonic graphoidal cover of a graph $G$ is a collection $psi_{dm}$ of detour monophonic paths in $G$ such that every vertex of $G$ is an internal vertex  of at most on...

We study the parameterized complexity of the classical Edge Hamiltonian Path problem and give several fixed-parameter tractability results. First, we settle an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT parameterized by vertex cover, and that it also admits a cubic kernel. We then show fixed-parameter tractability even for a generalization of the problem to arb...

We study generating functions for the number of involutions of length n avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary permutation τ of length k. In several interesting cases these generating functions depend only on k and can be expressed via Chebyshev polynomials of the second kind. In particular, we show that involutions of length n avoiding ...

The strong embedding conjecture states that every 2-connected graph has a closed 2-cell embedding in some surface , i . e . an embedding that each face is bounded by a circuit in the graph . A graph is called k -crosscap embeddable if it can be embedded in the surface of non-orientable genus k . We confirm the strong embedding conjecture for 5-crosscap embeddable graphs . As a corollary , every...

The precise form of tensorial transformations acting on a given collection of infinite matrices into another ; for such classical ideas connected with the summability field of double gai sequence spaces. In this paper the results are impose conditions on the tensor g so that it becomes a tensorial transformations from the metric space χ to the metric space C Keywords—tensorial transformations,d...