57 research outputs found
Covering line graphs with equivalence relations
An equivalence graph is a disjoint union of cliques, and the equivalence
number of a graph is the minimum number of equivalence
subgraphs needed to cover the edges of . We consider the equivalence number
of a line graph, giving improved upper and lower bounds: . This disproves a
recent conjecture that is at most three for triangle-free
; indeed it can be arbitrarily large.
To bound we bound the closely-related invariant
, which is the minimum number of orientations of such that for
any two edges incident to some vertex , both and are oriented
out of in some orientation. When is triangle-free,
. We prove that even when is triangle-free, it
is NP-complete to decide whether or not .Comment: 10 pages, submitted in July 200
On k-Equivalence Domination in Graphs
Let G = (V,E) be a graph. A subset S of V is called an equivalence set if every component of the induced subgraph (S) is complete. If further at least one component of (V − S) is not complete, then S is called a Smarandachely equivalence set
Split graphs and Block Representations
In this paper, we study split graphs and related classes of graphs from the
perspective of their sequence of vertex degrees and an associated lattice under
majorization. Following the work of Merris in 2003, we define blocks
, where is the degree sequence of a graph, and
and are sequences arising from . We use the
block representation to characterize membership in
each of the following classes: unbalanced split graphs, balanced split graphs,
pseudo-split graphs, and three kinds of Nordhaus-Gaddum graphs (defined by
Collins and Trenk in 2013). As in Merris' work, we form a poset under the
relation majorization in which the elements are the blocks
representing split graphs with a fixed number of
edges. We partition this poset in several interesting ways using what we call
amphoras, and prove upward and downward closure results for blocks arising from
different families of graphs. Finally, we show that the poset becomes a lattice
when a maximum and minimum element are added, and we prove properties of the
meet and join of two blocks.Comment: 23 pages, 7 Figures, 2 Table
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