The path partition number of a graph is the minimum number of paths required to partition the vertices. We consider upper bounds on the path partition number under minimum and maximum degree assumptions.

Let k ≥ j ≥ 1 be two integers, and letG be a simple graph such that j(δ(G)+1) ≥ k, where δ(G) is the minimum degree of G. A (j, k)-dominating function of a graph G is a function f from the vertex set V (G) to the set {0, 1, 2, . . . , j} such that for any vertex v ∈ V (G), the condition ∑ u∈N[v] f(u) ≥ k is fulfilled, where N [v] is the closed neighborhood of v. A set {f1, f2, . . . , fd} of (j...

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {±1,±2, . . . ,±k} is said to be a signed star {k}-dominating function on G if ∑ e∈E(v) f(e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. The signed star {k}-domination number of a graph G is γ{k}SS(G) = min{ ∑ e∈E f(e) | f is a S...

The barbiturates tested in this work (barbital, phenobarbital, thiopental and pentobarbital) modify the partition of synaptosomes in a Dextran T500-poly(ethylene glycol) 4000 two-phase system. Under adequate experimental conditions, the drugs increase the partition into the upper phase and this effect appears to be due to an action on the biological material and not on the interface potential o...

We study the problem of finding the smallest power of an input graph that has k disjoint dominating sets, where the ith power of an input graph G is constructed by adding edges between pairs of vertices in G at distance i or less, and a subset of vertices in a graph G is a dominating set if and only if every vertex in G is adjacent to a vertex in this subset. The problem is a different view of ...

Let G be a finite and simple graph with vertex set V (G), and let f: V (G)→ {−1, 1} be a two-valued function. If k > 1 is an integer and ∑ x∈N[v] f(x) > k for each v ∈ V (G), where N [v] is the closed neighborhood of v, then f is a signed k-dominating function on G. A set {f1, f2, . . . , fd} of signed kdominating functions on G with the property that ∑ d i=1 fi(x) 6 k for each x ∈ V (G), is ca...

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arcset $A(D)$. A twin signed total Roman dominating function (TSTRDF) on thedigraph $D$ is a function $f:V(D)rightarrow{-1,1,2}$ satisfyingthe conditions that (i) $sum_{xin N^-(v)}f(x)ge 1$ and$sum_{xin N^+(v)}f(x)ge 1$ for each $vin V(D)$, where $N^-(v)$(resp. $N^+(v)$) consists of all in-neighbors (resp.out-neighbors) of $v$, and (...