21 research outputs found
Induced Saturation Number
In this paper, we discuss a generalization of the notion of saturation in
graphs in order to deal with induced structures. In particular, we define , which is the fewest number of gray edges in a trigraph so that
no realization of that trigraph has an induced copy of , but changing any
white or black edge to gray results in some realization that does have an
induced copy of .
We give some general and basic results and then prove that for where is the path on 4
vertices. We also show how induced saturation in this setting extends to a
natural notion of saturation in the context of general Boolean formulas.Comment: 14 pages, 7 figure
Removing induced powers of cycles from a graph via fewest edits
What is the minimum proportion of edges which must be added to or removed
from a graph of density to eliminate all induced cycles of length ? The
maximum of this quantity over all graphs of density is measured by the edit
distance function, , a function which provides
a natural metric between graphs and hereditary properties.
Martin determined for all
when and determined
for . Peck determined
for all for odd cycles, and for
for even cycles. In this paper, we fully
determine the edit distance function for and . Furthermore, we
improve on the result of Peck for even cycles, by determining
for all ,
where for a constant . More generally, if is the
-th power of the cycle , we determine
for all in the case when
, thus improving on earlier work of Berikkyzy, Martin and Peck.Comment: 17 page