5,508 research outputs found
Partitioning a graph into highly connected subgraphs
Given , a -proper partition of a graph is a partition
of such that each part of induces a
-connected subgraph of . We prove that if is a graph of order
such that , then has a -proper partition with at
most parts. The bounds on the number of parts and the minimum
degree are both best possible. We then prove that If is a graph of order
with minimum degree , where
, then has a -proper partition into at most
parts. This improves a result of Ferrara, Magnant and
Wenger [Conditions for Families of Disjoint -connected Subgraphs in a Graph,
Discrete Math. 313 (2013), 760--764] and both the degree condition and the
number of parts are best possible up to the constant
Induced Subgraphs of Johnson Graphs
The Johnson graph J(n,N) is defined as the graph whose vertices are the
n-subsets of the set {1,2,...,N}, where two vertices are adjacent if they share
exactly n - 1 elements. Unlike Johnson graphs, induced subgraphs of Johnson
graphs (JIS for short) do not seem to have been studied before. We give some
necessary conditions and some sufficient conditions for a graph to be JIS,
including: in a JIS graph, any two maximal cliques share at most two vertices;
all trees, cycles, and complete graphs are JIS; disjoint unions and Cartesian
products of JIS graphs are JIS; every JIS graph of order n is an induced
subgraph of J(m,2n) for some m <= n. This last result gives an algorithm for
deciding if a graph is JIS. We also show that all JIS graphs are edge move
distance graphs, but not vice versa.Comment: 12 pages, 4 figure
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
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