91 research outputs found
Heavy subgraphs, stability and hamiltonicity
Let be a graph. Adopting the terminology of Broersma et al. and \v{C}ada,
respectively, we say that is 2-heavy if every induced claw () of
contains two end-vertices each one has degree at least ; and
is o-heavy if every induced claw of contains two end-vertices with degree
sum at least in . In this paper, we introduce a new concept, and
say that is \emph{-c-heavy} if for a given graph and every induced
subgraph of isomorphic to and every maximal clique of ,
every non-trivial component of contains a vertex of degree at least
in . In terms of this concept, our original motivation that a
theorem of Hu in 1999 can be stated as every 2-connected 2-heavy and
-c-heavy graph is hamiltonian, where is the graph obtained from a
triangle by adding three disjoint pendant edges. In this paper, we will
characterize all connected graphs such that every 2-connected o-heavy and
-c-heavy graph is hamiltonian. Our work results in a different proof of a
stronger version of Hu's theorem. Furthermore, our main result improves or
extends several previous results.Comment: 21 pages, 6 figures, finial version for publication in Discussiones
Mathematicae Graph Theor
Packing 3-vertex paths in claw-free graphs and related topics
An L-factor of a graph G is a spanning subgraph of G whose every component is
a 3-vertex path. Let v(G) be the number of vertices of G and d(G) the
domination number of G. A claw is a graph with four vertices and three edges
incident to the same vertex. A graph is claw-free if it has no induced subgraph
isomorphic to a claw. Our results include the following. Let G be a 3-connected
claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in
G. Then
(a1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (a2)
if v(G) = 1 mod 3, then G - x has an L-factor, (a3) if v(G) = 2 mod 3, then G -
{x,y} has an L-factor, (a4) if v(G) = 0 mod 3 and G is either cubic or
4-connected, then G - P has an L-factor, (a5) if G is cubic with v(G) > 5 and E
is a set of three edges in G, then G - E has an L-factor if and only if the
subgraph induced by E in G is not a claw and not a triangle, (a6) if v(G) = 1
mod 3, then G - {v,e} has an L-factor for every vertex v and every edge e in G,
(a7) if v(G) = 1 mod 3, then there exist a 4-vertex path N and a claw Y in G
such that G - N and G - Y have L-factors, and (a8) d(G) < v(G)/3 +1 and if in
addition G is not a cycle and v(G) = 1 mod 3, then d(G) < v(G)/3.
We explore the relations between packing problems of a graph and its line
graph to obtain some results on different types of packings. We also discuss
relations between L-packing and domination problems as well as between induced
L-packings and the Hadwiger conjecture.
Keywords: claw-free graph, cubic graph, vertex disjoint packing, edge
disjoint packing, 3-vertex factor, 3-vertex packing, path-factor, induced
packing, graph domination, graph minor, the Hadwiger conjecture.Comment: 29 page
Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms
An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both.
Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results
Forbidden subgraphs that imply Hamiltonian-connectedness
It is proven that if is a -connected claw-free graph which is also -free (where is a triangle with a path of length attached), -free (where is a path with vertices) or -free (where consists of two disjoint triangles connected by an edge), then is Hamiltonian-connected. Also, examples will be described that determine a finite family of graphs such that if a 3-connected graph being claw-free and -free implies is Hamiltonian-connected, then . \u
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