A total Roman dominating function on a graph
is a function
such that every
vertex
with
is adjacent to
some vertex
with
, and the subgraph of
induced by the set
of all vertices
such
that
has no isolated
vertices. The weight of
is
. The total Roman
domination number
is the minimum weight of a total Roman dominating function on
. A graph
is
--edge-critical
if
for every
edge
, and
--edge-supercritical
if it is
--edge-critical
and
for every edge
. We present some basic
results on
-edge-critical
graphs and characterize certain classes of
-edge-critical graphs. In
addition, we show that, when
is small, there is a connection between
--edge-critical
graphs and graphs which are critical with respect to the domination and total
domination numbers.
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