4,534 research outputs found
Countable connected-homogeneous digraphs
A digraph is connected-homogeneous if every isomorphism between two finite
connected induced subdigraphs extends to an automorphism of the whole digraph.
In this paper, we completely classify the countable connected-homogeneous
digraphs.Comment: 49 page
Structure of Cubic Lehman Matrices
A pair of square -matrices is called a \emph{Lehman pair} if
for some integer . In this case and
are called \emph{Lehman matrices}. This terminology arises because Lehman
showed that the rows with the fewest ones in any non-degenerate minimally
nonideal (mni) matrix form a square Lehman submatrix of . Lehman
matrices with are essentially equivalent to \emph{partitionable graphs}
(also known as -graphs), so have been heavily studied as part
of attempts to directly classify minimal imperfect graphs. In this paper, we
view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite
graph, focusing in particular on the case where the graph is cubic. From this
perspective, we identify two constructions that generate cubic Lehman graphs
from smaller Lehman graphs. The most prolific of these constructions involves
repeatedly replacing suitable pairs of edges with a particular -vertex
subgraph that we call a -rung ladder segment. Two decades ago, L\"{u}tolf \&
Margot initiated a computational study of mni matrices and constructed a
catalogue containing (among other things) a listing of all cubic Lehman
matrices with of order up to . We verify their catalogue
(which has just one omission), and extend the computational results to matrices. Of the cubic Lehman matrices (with ) of order
up to , only two do not arise from our -rung ladder
construction. However these exceptions can be derived from our second
construction, and so our two constructions cover all known cubic Lehman
matrices with
On the size of identifying codes in triangle-free graphs
In an undirected graph , a subset such that is a
dominating set of , and each vertex in is dominated by a distinct
subset of vertices from , is called an identifying code of . The concept
of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in
1998. For a given identifiable graph , let \M(G) be the minimum
cardinality of an identifying code in . In this paper, we show that for any
connected identifiable triangle-free graph on vertices having maximum
degree , \M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}. This bound is
asymptotically tight up to constants due to various classes of graphs including
-ary trees, which are known to have their minimum identifying code
of size . We also provide improved bounds for
restricted subfamilies of triangle-free graphs, and conjecture that there
exists some constant such that the bound \M(G)\le n-\tfrac{n}{\Delta}+c
holds for any nontrivial connected identifiable graph
Centroidal bases in graphs
We introduce the notion of a centroidal locating set of a graph , that is,
a set of vertices such that all vertices in are uniquely determined by
their relative distances to the vertices of . A centroidal locating set of
of minimum size is called a centroidal basis, and its size is the
centroidal dimension . This notion, which is related to previous
concepts, gives a new way of identifying the vertices of a graph. The
centroidal dimension of a graph is lower- and upper-bounded by the metric
dimension and twice the location-domination number of , respectively. The
latter two parameters are standard and well-studied notions in the field of
graph identification.
We show that for any graph with vertices and maximum degree at
least~2, . We discuss the
tightness of these bounds and in particular, we characterize the set of graphs
reaching the upper bound. We then show that for graphs in which every pair of
vertices is connected via a bounded number of paths,
, the bound being tight for paths and
cycles. We finally investigate the computational complexity of determining
for an input graph , showing that the problem is hard and cannot
even be approximated efficiently up to a factor of . We also give an
-approximation algorithm
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