5,157 research outputs found
Forbidden Subgraphs in Connected Graphs
Given a set of connected non acyclic graphs, a
-free graph is one which does not contain any member of as copy.
Define the excess of a graph as the difference between its number of edges and
its number of vertices. Let {\gr{W}}_{k,\xi} be theexponential generating
function (EGF for brief) of connected -free graphs of excess equal to
(). For each fixed , a fundamental differential recurrence
satisfied by the EGFs {\gr{W}}_{k,\xi} is derived. We give methods on how to
solve this nonlinear recurrence for the first few values of by means of
graph surgery. We also show that for any finite collection of non-acyclic
graphs, the EGFs {\gr{W}}_{k,\xi} are always rational functions of the
generating function, , of Cayley's rooted (non-planar) labelled trees. From
this, we prove that almost all connected graphs with nodes and edges
are -free, whenever and by means of
Wright's inequalities and saddle point method. Limiting distributions are
derived for sparse connected -free components that are present when a
random graph on nodes has approximately edges. In particular,
the probability distribution that it consists of trees, unicyclic components,
, -cyclic components all -free is derived. Similar results are
also obtained for multigraphs, which are graphs where self-loops and
multiple-edges are allowed
Ore- and Fan-type heavy subgraphs for Hamiltonicity of 2-connected graphs
Bedrossian characterized all pairs of forbidden subgraphs for a 2-connected
graph to be Hamiltonian. Instead of forbidding some induced subgraphs, we relax
the conditions for graphs to be Hamiltonian by restricting Ore- and Fan-type
degree conditions on these induced subgraphs. Let be a graph on
vertices and be an induced subgraph of . is called \emph{o}-heavy if
there are two nonadjacent vertices in with degree sum at least , and is
called -heavy if for every two vertices ,
implies that . We say that is -\emph{o}-heavy
(-\emph{f}-heavy) if every induced subgraph of isomorphic to is
\emph{o}-heavy (\emph{f}-heavy). In this paper we characterize all connected
graphs and other than such that every 2-connected
-\emph{f}-heavy and -\emph{f}-heavy (-\emph{o}-heavy and
-\emph{f}-heavy, -\emph{f}-heavy and -free) graph is Hamiltonian. Our
results extend several previous theorems on forbidden subgraph conditions and
heavy subgraph conditions for Hamiltonicity of 2-connected graphs.Comment: 21 pages, 2 figure
Forbidden triples and traceability: a characterization
AbstractGiven a connected graph G, a family F of connected graphs is called a forbidden family if no induced subgraph of G is isomorphic to any graph in F. If this is the case, G is said to be F-free. In earlier papers the authors identified four distinct families of triples of subgraphs that imply traceability when they are forbidden in sufficiently large graphs. In this paper the authors introduce a fifth family and show these are all such families
Forbidden induced subgraphs and the price of connectivity for feedback vertex set.
Let fvs(G) and cfvs(G) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of a graph G, respectively. For a graph class G, the price of connectivity for feedback vertex set (poc-fvs) for G is defined as the maximum ratio cfvs(G)/fvs(G) over all connected graphs G in G. It is known that the poc-fvs for general graphs is unbounded. We study the poc-fvs for graph classes defined by a finite family H of forbidden induced subgraphs. We characterize exactly those finite families H for which the poc-fvs for H-free graphs is bounded by a constant. Prior to our work, such a result was only known for the case where |H|=1
A pair of forbidden subgraphs and perfect matchings
AbstractIn this paper, we study the relationship between forbidden subgraphs and the existence of a matching. Let H be a set of connected graphs, each of which has three or more vertices. A graph G is said to be H-free if no graph in H is an induced subgraph of G. We completely characterize the set H such that every connected H-free graph of sufficiently large even order has a perfect matching in the following cases.(1)Every graph in H is triangle-free.(2)H consists of two graphs (i.e. a pair of forbidden subgraphs).A matching M in a graph of odd order is said to be a near-perfect matching if every vertex of G but one is incident with an edge of M. We also characterize H such that every H-free graph of sufficiently large odd order has a near-perfect matching in the above cases
Recommended from our members
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
- …