New bounds on the signed total domination number of graphs

Abstract

In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turán [8], we present a sharp lower bound on Kr+1-free graphs for r ≥ 2. Applying the concept of total limited packing we bound the signed total domination number of G with δ(G) ≥ 3 from above by n − 2 2ρo(G)+δ−3 2 . Also, we prove that γst(T ) ≤ n − 2(s − s ) for any tree T of order n, with s support vertices and s support vertices of degree two. Moreover, we characterize all trees attaining this bound.

On domination-type invariants of Fibonacci cubes and hypercubes

“…The signed domination number γ s (G) is the minimum of v∈V (G) f (v) taken over all signed dominating functions f of G, see [18].…”

On the inverse signed total domination number in graphs

“…As an immediate result of Theorem 3.1 we have γ 0 st (G) = −γ st (G), for all cubic graph G. Hosseini Moghaddam et al [3] showed that γ st (G) ≤ 2n/3 is a sharp upper bound for all connected cubic graph G different from the Heawood graph G 14 . Therefore, if G is a connected cubic graph different from G 14 , then γ 0 st (G) ≥ −2n/3 is a sharp lower bound.…”

On the $k$-limited packing numbers in graphs

“…Here, we prefer to work with these definitions on these parameters rather than the previous ones. Gallant et al [5] introduced the concept of limited packing in graphs and exhibited some real-world applications of it to network security, market saturation and codes (Also, the authors in [7] presented some results as an application of the concept of the limited packing). A subset…”