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Locating-total dominating sets in twin-free graphs: a conjecture
A total dominating set of a graph is a set of vertices of such
that every vertex of has a neighbor in . A locating-total dominating set
of is a total dominating set of with the additional property that
every two distinct vertices outside have distinct neighbors in ; that
is, for distinct vertices and outside , where denotes the open neighborhood of . A graph is twin-free if
every two distinct vertices have distinct open and closed neighborhoods. The
location-total domination number of , denoted , is the minimum
cardinality of a locating-total dominating set in . It is well-known that
every connected graph of order has a total dominating set of size at
most . We conjecture that if is a twin-free graph of order
with no isolated vertex, then . We prove the
conjecture for graphs without -cycles as a subgraph. We also prove that if
is a twin-free graph of order , then .Comment: 18 pages, 1 figur
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