32 research outputs found
On the λ'-optimality of s-geodetic digraphs
For a strongly connected digraph D the restricted arc-connectivity λ'(D) is defined as the minimum cardinality of an arc-cut over all arc-cuts S satisfying that D − S has a non trivial strong component D1 such that D − V (D1) contains an arc. Let S be a subset of vertices of D. We denote by ω+(S) the set of arcs uv with u ∈ S and v ∈ S, and by ω−(S) the set of arcs uv with
u ∈ S and v ∈ S. A digraph D = (V,A) is said to be λ'-optimal if λ'(D) = ξ'(D), where ξ'(D) is the minimum arc-degree of D defined as ξ(D) = min{ξ'(xy) : xy ∈ A}, and ξ'(xy) = min{|ω+({x, y})|, |ω−({x, y})|, |ω+(x)∪ω−(y)|, |ω−(x)∪ω+(y)|}.
In this paper a sufficient condition for a s-geodetic strongly connected digraph D to be λ'-optimal is given in terms of its diameter.Further we see that the h-iterated line digraph Lh(D) of a s-geodetic digraph is λ'-optimal for certain iteration h.Peer Reviewe
Disconnecting strongly regular graphs
In this paper, we show that the minimum number of vertices whose removal
disconnects a connected strongly regular graph into non-singleton components,
equals the size of the neighborhood of an edge for many graphs. These include
blocks graphs of Steiner -designs, many Latin square graphs and strongly
regular graphs whose intersection parameters are at most a quarter of their
valency
Extra Connectivity of Strong Product of Graphs
The - of a connected graph is
the minimum cardinality of a set of vertices, if it exists, whose deletion
makes disconnected and leaves each remaining component with more than
vertices, where is a non-negative integer. The of graphs and is the graph with vertex set , where two distinct vertices are adjacent in if and only if and or
and or and . In this paper, we give the - of
, where is a maximally connected -regular graph for . As a byproduct, we get -
conditional fault-diagnosability of under model
On the complexity of computing the -restricted edge-connectivity of a graph
The \emph{-restricted edge-connectivity} of a graph , denoted by
, is defined as the minimum size of an edge set whose removal
leaves exactly two connected components each containing at least vertices.
This graph invariant, which can be seen as a generalization of a minimum
edge-cut, has been extensively studied from a combinatorial point of view.
However, very little is known about the complexity of computing .
Very recently, in the parameterized complexity community the notion of
\emph{good edge separation} of a graph has been defined, which happens to be
essentially the same as the -restricted edge-connectivity. Motivated by the
relevance of this invariant from both combinatorial and algorithmic points of
view, in this article we initiate a systematic study of its computational
complexity, with special emphasis on its parameterized complexity for several
choices of the parameters. We provide a number of NP-hardness and W[1]-hardness
results, as well as FPT-algorithms.Comment: 16 pages, 4 figure
Further topics in connectivity
Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered.
For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version