Working on the problem of finding the numbers of lattice points inside convex lattice polygons, Rabinowitz has made several conjectures dealing with convex lattice nonagons and decagons. An intensive computer search preceded a formulation of the conjectures. The main purpose of this paper is to prove some of Rabinowitz’s conjectures. Moreover, we obtain an improvement of a conjectured result and give short proofs of two known results.

Let \(k \geq 1\) be an integer and let \(G = (V, E)\) be a graph. A set \(S\) of vertices of \(G\) is \(k\)-independent if the distance between any two vertices of \(S\) is at least \(k+1\). We denote by \(\rho_k(G)\) the maximum cardinality among all \(k\)-independent sets of \(G\). Number \(\rho_k(G)\) is called the \(k\)-packing number of \(G\). Furthermore, \(S\) is defined to be \(k\)-dominating set in \(G\) if every vertex in \(V(G) – S\) is at distance at most \(k\) from some vertex in \(S\). A set \(S\) is \(k\)-independent dominating if it is both \(k\)-independent and \(k\)-dominating. The \(k\)-independent dominating number, \(i_k(G)\), is the minimum cardinality among all \(k\)-independent dominating sets of \(G\). We find the values \(i_k(G)\) and \(\rho_k(G)\) for iterated line graphs.

The maximum genus, a topological invariant of graphs, was inaugurated by Nordhaus \(et\; al\). \([16]\). In this paper, the relations between the maximum non-adjacent edge set and the upper embeddability of a graph are discussed, and the lower bounds on maximum genus of a graph in terms of its girth and maximum non-adjacent edge set are given. Furthermore, these bounds are shown to be best possible. Thus, some new results on the upper embeddability and the lower bounds on the maximum genus of graphs are given.

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well known vertex covering and dominating set problems in graphs (see SIAM J. Discrete Math. \(15(4) (2002), 519-529)\). A set \(S\) of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set \(S\) (following a set of rules for power system monitoring). The minimum cardinality of a power dominating set of a graph is its power domination number. We investigate the power domination number of a block graph.

A \((p,q)\)-graph \(G\) in which the edges are labeled \(1,2,3,\ldots,q\) so that the vertex sums are constant, is called supermagic. If the vertex sum mod \(p\) is a constant, then \(G\) is called edge-magic. We investigate the supermagic characteristic of a simple graph \(G\), and its edge-splitting extension \(SPE(G,f)\). The construction provides an abundance of new supermagic multigraphs.

The basis number of a graph \(G\) is defined to be the least integer \(k\) such that \(G\) has a \(k\)-fold basis for its cycle space. We investigate the basis number of the composition of theta graphs, a theta graph and a path, a theta graph and a cycle, a path and a theta graph, and a cycle and a theta graph.

We introduce certain types of surfaces \(M_j^n\), for \(j = 1,2,\ldots,11\) and determine their genus distributions. At the basis of joint trees introduced by Liu, we develop the surface sorting method to calculate the embedding distribution by genus.

Network reliability is an important issue in the area of distributed computing. Most of the early work in this area takes a probabilistic approach to the problem. However, sometimes it is important to incorporate subjective reliability estimates into the measure. To serve this goal, we propose the use of the weighted integrity, a measure of graph vulnerability. The weighted integrity problem is known to be NP-complete for most of the common network topologies including tree, mesh, hypercube, etc. It is known to be NP-complete even for most perfect graphs, including comparability graphs and chordal graphs. However, the computational complexity of the problem is not known for one class of perfect graphs, namely, co-comparability graphs. In this paper, we give a polynomial-time algorithm to compute the weighted integrity of interval graphs, a subclass of co-comparability graphs.

The vertex linear arboricity \(vla(G)\) of a graph \(G\) is the minimum number of subsets into which the vertex set \(V(G)\) can be partitioned so that each subset induces a subgraph whose connected components are paths. An integer distance graph is a graph \(G(D)\) with the set of all integers as vertex set and two vertices \(u,v \in {Z}\) are adjacent if and only if \(|u-v| \in D\) where the distance set \(D\) is a subset of the positive integers set. Let \(D_{m,k} = \{1,2,\ldots,m\} – \{k\}\) for \(m > k \geq 1\). In this paper, some upper and lower bounds of the vertex linear arboricity of the integer distance graph \(G(D_{m,k})\) are obtained. Moreover, \(vla(G(D_{m,1})) = \lceil \frac{m}{4} \rceil +1\) for \(m \geq 3\), \(vla(G(D_{8l+1,2})) = 2l + 2\) for any positive integer \(l\) and \(vla(G(D_{4q,2})) = q+2\) for any integer \(q \geq 2\).

We determine all spreads of symmetry of the dual polar space \(H^D(2n-1,q^2)\). We use this to show the existence of glued near polygons of type \(H^D(2n_1-1,q^2) \otimes H^D(2n_2-1,q^2)\). We also show that there exists a unique glued near polygon of type \(H^D(2n_1-1,4) \otimes H^D(2n_2-1,4)\) for all \(n_1,n_2 \geq 2\). The unique glued near polygon of type \(H^D(2n-1,4) \otimes Q(2n_2-1,q^2)\) has the property that it contains \(H^D(2n-1,4)\) as a big geodetically closed sub near polygon. We will determine all dense near \((2n+2)\)-gons, \(n \geq 3\), which have \(H^D(2n-1,4)\) as a big geodetically closed sub near polygon. We will prove that such a near polygon is isomorphic to either \(H^D(2n+1,4)\), \(H^D(2n-1,4) \otimes Q(5,2)\) or \(H^D(2n-1,4) \times L\) for some line \(L\) of size at least three.