For two vertices \(u\)  and \(v\) of a connected graph \(G\) , the set \(H(u,v)\) consists of all those vertices lying on a \(u-v\) geodesic in \(G\) . Given a set \(S\) of vertices of \(G\) , the union of all sets \(H(u,v)\) for \(u,v\in S\) is denoted by \(H(S)\) . A convex set \(S\) satisfies \(H(S)=S\) . The convex hull \([S]\) is the smallest convex set containing \(S\) . The hull number \(h(G)\) is the minimum cardinality among the subsets \(S\) of \(V(G)\) with \([S]=V(G)\) . A set \(S\) is a geodetic set if \(H(S)=V(G)\) ; while \(S\) is a hull set if \([S]=V(G)\) . The minimum cardinality of a geodetic set of \(G\) is the geodetic number \(g(G)\) . A subset \(T\) of a minimum hull set \(S\) is called a forcing subset for \(S\) if \(S\) is the unique minimum hull set containing \(T\) . The forcing hull number \(f(S,h)\) of \(S\) is the minimum cardinality among the forcing subsets of \(S\) , and the forcing hull number \(f(G,h)\) of \(G\) is the minimum forcing hull number among all minimum hull sets of \(G\) . The forcing geodetic number \(f(S,g)\)  of a minimum geodetic set \(S\) in \(G\) and the forcing geodetic number \(f(G,g)\) of \(G\) are defined in a similar fashion. The forcing hull numbers of several classes of graphs are determined. It is shown that for integers \(a,b\) with \(0\leq a\leq b\) , there exists a connected graph \(G\) such that \(f(G,h)=a\) and \(h(G)=b\) . We investigate the realizability of integers \(a,b\geq0\) that are the forcing hull and forcing geodetic numbers, respectively, of some graph.

Let \(C\) be a perfect 1-error-correcting code of length \(15\). We show that a quotient \(H(C)\) of the minimum distance graph of \(C\) constitutes an invariant for \(C\) more sensible than those studied up to the present, namely the kernel dimension and the rank. As a by-product, we get a nonlinear Vasil’ev code \(C\) all of whose associated Steiner triple systems are linear. Finally, the determination of \(H(C)\) for known families of \(C\)’s is presented.

A computer search shows that there does not exist a nested BIB design \(\text{NB}(10, 15, 2, 3)\).

We construct several families of simple 4-designs, which are closely related to Alltop’s series with parameters \(4-(2^f+1,5,5)\), \(f\) odd. More precisely, for every \(q=2^f\), where \(gcd(f,6)=1\), \(f\geq5\), we construct designs with the following parameters:

\[4-(q+1,6,\lambda),\, \text{where}\, \lambda\in\{60,70,90,100,150,160\},\]
\[4-(q+1,8,35),\]
\[4-(q+1,9,\lambda),\, \text{where}\, \lambda\in\{63,147\}.\]

Eulerian numbers may be defined recursively and have applications to many branches of mathematics. We derive some congruence and divisibility properties of Eulerian numbers.

In this paper, we determine the spectrum of support sizes of indecomposable threefold triple systems of order \(v\) for all \(v > 15\).