The forcing total monophonic number of a graph

For a connected graph G = (V, E) of order at least two, a subset T of a minimum total monophonic set S of G is a forcing total monophonic subset for S if S is the unique minimum total monophonic set containing T . A forcing total monophonic subset for S of minimum cardinality is a minimum forcing total monophonic subset of S. The forcing total monophonic number ftm(S) in G is the cardinality of a minimum forcing total monophonic subset of S. The forcing total monophonic number of G is ftm(G) = min{ftm(S)}, where the minimum is taken over all minimum total monophonic sets S in G. We determine bounds for it and find the forcing total monophonic number of certain classes of graphs. It is shown that for every pair a, b of positive integers with 0 ≤ a < b and b ≥ a+4, there exists a connected graph G such that ftm(G) = a and mt(G) = b.

The forcing near geodetic number of a graph

A set [Formula: see text] is a near geodetic set if for every [Formula: see text] in [Formula: see text] there exist some [Formula: see text] in [Formula: see text] with [Formula: see text] The near geodetic number [Formula: see text] is the minimum cardinality of a near geodetic set in [Formula: see text] A subset [Formula: see text] of a minimum near geodetic set [Formula: see text] is called the forcing subset of [Formula: see text] if [Formula: see text] is the unique minimum near geodetic set containing [Formula: see text]. The forcing number [Formula: see text] of [Formula: see text] in [Formula: see text] is the minimum cardinality of a forcing subset for [Formula: see text], while the forcing near geodetic number [Formula: see text] of [Formula: see text] is the smallest forcing number among all minimum near geodetic sets of [Formula: see text]. In this paper, we initiate the study of forcing near geodetic number of connected graphs. We characterize graphs with [Formula: see text]. Further, we compare the parameters geodetic number[Formula: see text] near geodetic number[Formula: see text] forcing near geodetic number and we proved that, for every positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], there exists a nontrivial connected graph [Formula: see text] with [Formula: see text] [Formula: see text] and [Formula: see text].

Trees with unique minimum glolal offensive alliance

Let G= (V,E) be a simple graph. A non-empty set D⊆V is called a global offensive alliance if D is a dominating set and for every vertex v in V-D, |N_{G}[v]∩D|≥|N_{G}[v]-D|. The global offensive alliance number is the minimum cardinality of a global offensive alliance in G. In this paper, we give a constructive characterization of trees having a unique minimum global offensive alliance.

The Forcing Restrained Steiner Number of a Graph

A restrained Steiner set of a connected graph 𝑮 of order 𝒑 ≥ 𝟐 is a set 𝑾 ⊆ 𝑽(𝑮)such that 𝑾 is a Steiner set, and if either 𝑾 = 𝑽 or the subgraph𝑮[𝑽 − 𝑾] inducedby [𝑽 − 𝑾] has no isolated vertices. The restrained Steiner number 𝒔𝒓 𝑮 of 𝑮 isthe minimum cardinality of its restrained Steiner sets and any restrained Steinerset of cardinality 𝒔𝒓 𝑮 is a minimum restrained Steiner set of 𝑮. For a minimum restrained Steiner set 𝑾of 𝑮, a subset 𝑻 ⊆ 𝑾 is called a forcing subset for 𝑾 if 𝑾is the unique minimum restrained Steiner set containing 𝑻. A forcing subset for 𝑾of minimum cardinality is a minimum forcing subset of 𝑾. The forcing restrained Steiner number of 𝑾, denoted by 𝒇𝒓𝒔 𝑾 , is the cardinality of a minimum forcingsubset of 𝑾. The forcing restrained Steiner number of 𝑮, denoted by 𝒇𝒓𝒔 𝑮 is𝒇𝒓𝒔 𝑮 = 𝒎𝒊𝒏⁡{𝒇𝒓𝒔 𝑾 }, where the minimum is taken over all minimum restrainedSteiner sets 𝑾 in 𝑮. Some general properties satisfied by the concept forcing restrained Steiner number are studied. The forcing restrained Steiner number of certain classes of graphs is determined. It is shown that for every pair 𝒂,𝒃 ofintegers with 𝟎 ≤ 𝒂 < 𝒃and 𝒃 ≥ 𝟐, there exists a connected graph 𝑮 such that𝒇𝒓𝒔 𝑮 = 𝒂 and 𝒔𝒓 𝑮 = 𝒃.

Quinn’s Law of Fluid Dynamics Pressure-Driven Fluid Flow through Closed Conduits

In this paper we develop from first principles a unique law pertaining to the flow of fluids through closed conduits. This law, which we call &ldquo;Quinn&rsquo;s Law&rdquo;, may be described as follows: When fluids are forced to flow through closed conduits under the driving force of a pressure gradient, there is a linear relationship between the normalized dimensionless pressure gradient, PQ, and the normalized dimensionless fluid current, CQ. The relationship is expressed mathematically as: PQ = k1 +k2CQ. This linear relationship remains the same whether the conduit is filled with or devoid of solid obstacles. The law differentiates, however, between a packed and an empty conduit by virtue of the tortuosity of the fluid path, which is seamlessly accommodated within the normalization framework of the law itself. When movement of the fluid is very close to being at rest, i.e., very slow, this relationship has the unique minimum constant value of k1, and as the fluid acceleration increases, it varies with a slope of k2 as a function of normalized fluid current. Quinn&rsquo;s Law is validated herein by applying it to the data from published classical studies of measured permeability in both packed and empty conduits, as well as to the data generated by home grown experiments performed in the author&rsquo;s own laboratory.

The cd-Indices of Intervals in the Uncrossing Partial Order on Matchings

We study flag enumeration in intervals in the uncrossing partial order on matchings. We produce a recursion for the cd-indices of intervals in the uncrossing poset $P_n$. We explicitly describe the matchings by constructing an order-reversing bijection. We obtain a recursion for the ab-indices of intervals in the poset $\hat{P}_n$, the poset $P_n$ with a unique minimum $\hat0$ adjoined.

The Semigroup of Surjective Transformations on an Infinite Set

For an infinite set X, denote by Ω(X) the semigroup of all surjective mappings from X to X. We determine Green’s relations in Ω(X), show that the kernel (unique minimum ideal) of Ω(X) exists and determine its elements and cardinality. For a countably infinite set X, we describe the elements of Ω(X) for which the 𝒟-class and 𝒥-class coincide. We compare the results for Ω(X) with the corresponding results for other transformation semigroups on X.