283 research outputs found
Minimal Connectivity
A k-connected graph such that deleting any edge / deleting any vertex /
contracting any edge results in a graph which is not k-connected is called
minimally / critically / contraction-critically k-connected. These three
classes play a prominent role in graph connectivity theory, and we give a brief
introduction with a light emphasis on reduction- and construction theorems for
classes of k-connected graphs.Comment: IMADA-preprint-math, 33 page
A degree and forbidden subgraph condition for a k-contractible edge
An edge in a k-conected graph is said to be k-contractible if the contraction of it results in a k-connected graph. We say that k-connected graph G satis es “ degree-sum conditon ”if Σx2V (W)degG(x) 3k +2 holds for any connected subgraph W of G with |W|= 3. Let k be an integer such that k 5. We prove that if a k-connected graph with no K1+C4 satis es degree-sum condition, then it has a k-contractible edge.電気通信大学201
Defining Recursive Predicates in Graph Orders
We study the first order theory of structures over graphs i.e. structures of
the form () where is the set of all
(isomorphism types of) finite undirected graphs and some vocabulary. We
define the notion of a recursive predicate over graphs using Turing Machine
recognizable string encodings of graphs. We also define the notion of an
arithmetical relation over graphs using a total order on the set
such that () is isomorphic to
().
We introduce the notion of a \textit{capable} structure over graphs, which is
one satisfying the conditions : (1) definability of arithmetic, (2)
definability of cardinality of a graph, and (3) definability of two particular
graph predicates related to vertex labellings of graphs. We then show any
capable structure can define every arithmetical predicate over graphs. As a
corollary, any capable structure also defines every recursive graph relation.
We identify capable structures which are expansions of graph orders, which are
structures of the form () where is a partial order. We
show that the subgraph order i.e. (), induced subgraph
order with one constant i.e. () and an expansion
of the minor order for counting edges i.e. ()
are capable structures. In the course of the proof, we show the definability of
several natural graph theoretic predicates in the subgraph order which may be
of independent interest. We discuss the implications of our results and
connections to Descriptive Complexity
Rearrangement Groups of Fractals
We construct rearrangement groups for edge replacement systems, an infinite
class of groups that generalize Richard Thompson's groups F, T, and V .
Rearrangement groups act by piecewise-defined homeomorphisms on many
self-similar topological spaces, among them the Vicsek fractal and many Julia
sets. We show that every rearrangement group acts properly on a locally finite
CAT(0) cubical complex, and we use this action to prove that certain
rearrangement groups are of type F infinity.Comment: 48 pages, 37 figure
Cographs and 1-sums
A graph that can be generated from using joins and 0-sums is called a
cograph. We define a sesquicograph to be a graph that can be generated from
using joins, 0-sums, and 1-sums. We show that, like cographs,
sesquicographs are closed under induced minors. Cographs are precisely the
graphs that do not have the 4-vertex path as an induced subgraph. We obtain an
analogue of this result for sesquicographs, that is, we find those
non-sesquicographs for which every proper induced subgraph is a sesquicograph
Edge Contraction and Line Graphs
Given a family of graphs , a graph is -free if
any subset of does not induce a subgraph of that is isomorphic to
any graph in . We present sufficient and necessary conditions for
a graph such that is -free for any edge in .
Thereafter, we use these conditions to characterize claw-free and line graphs.Comment: arXiv admin note: text overlap with arXiv:2203.0349
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