Edge even graceful labeling of a graph
G
with
p
vertices and
q
edges is a bijective
f
from the set of edge
E
G
to the set of positive integers
2,4
,
…
,
2
q
such that all the vertex labels
f
∗
V
G
, given by
f
∗
u
=
∑
u
v
∈
E
G
f
u
v
mod
2
k
, where
k
=
max
p
,
q
, are pairwise distinct. There are many graphs that do not have edge even graceful labeling, so in this paper, we have extended the definition of edge even graceful labeling to
r
-edge even graceful labeling and strong
r
-edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an
r
-edge even graceful graph. Furthermore, the minimum number
r
for which the graphs: tortoise graph, double star graph, ladder and diagonal ladder graphs, helm graph, crown graph, sunflower graph, and sunflower planar graph, have an
r
-edge even graceful labeling was found. Finally, we proved that the even cycle
C
2
n
has a strong
2
-edge even graceful labeling when
n
is even.