Rotation distance between rooted binary trees is the minimum number of simple rotations needed to transform one tree into the other. Computing the rotation distance between a pair of rooted trees can be quickly reduced in cases where there is a common edge between the trees, or where a single rotation introduces a common edge. Tree pairs which do not have such a reduction are difficult tree pairs, where there is no generally known first step. Here, we describe efforts to count and estimate the number of such difficult tree pairs, and find that the fraction decreases exponentially fast toward zero. We also describe how knowing the number of distinct instances of the rotation distance problem is a helpful factor in making the computations more feasible.

In this paper, we characterize the set of spanning trees of \( G^1_{n,r} \) (a simple connected graph consisting of \( n \) edges, containing exactly one \textit{1-edge-connected chain} of \( r \) cycles \( C^1_r \) and \( G^1_{n,r} \), where \( C^1_r \) is a \textit{forest}). We compute the Hilbert series of the face ring \( k[\Delta_s(G^1_{n,r})] \) for the spanning simplicial complex \( \Delta_s(G^1_{n,r}) \). Also, we characterize associated primes of the facet ideal \( I_F(\Delta_s(G^1_{n,r})) \). Furthermore, we prove that the face ring \( k[\Delta_s(G^1_{n,r})] \) is Cohen-Macaulay.

Unlike undirected graphs where the concept of Roman domination has been studied very extensively over the past 15 years, Roman domination remains little studied in digraphs. However, the published works are quite varied and deal with different aspects of Roman domination, including for example, Roman bondage, Roman reinforcement, Roman dominating family of functions and the signed version of some Roman dominating functions. In this survey, we will explore some Roman domination related results on digraphs, some of which are extensions of well-known properties of the Roman domination parameters of undirected graphs.

Let \( K_{g_1,g_2,\dots,g_n} \) be a complete \( n \)-partite graph with partite sets of sizes \( g_i \) for \( 1 \leq i \leq n \). A complete \( n \)-partite is balanced if each partite set has \( g \) vertices. In this paper, we will solve the problem of finding a maximum packing of the balanced complete \( n \)-partite graph, \( n \) even, with edge-disjoint 5-cycles when the leave is a 1-factor.

A double Italian dominating function on a graph \( G \) with vertex set \( V(G) \) is defined as a function \( f : V(G) \to \{0,1,2,3\} \) such that each vertex \( u \in V(G) \) with \( f(u) \in \{0,1\} \) has the property that \( \sum_{x \in N[u]} f(x) \geq 3 \), where \( N[u] \) is the closed neighborhood of \( u \). A set \( \{f_1, f_2, \dots, f_d\} \) of distinct double Italian dominating functions on \( G \) with the property that \( \sum_{i=1}^{d} f_i(v) \leq 3 \) for each \( v \in V(G) \) is called a \textit{double Italian dominating family} (of functions) on \( G \). The maximum number of functions in a double Italian dominating family on \( G \) is the double Italian domatic number of \( G \), denoted by \( dd_I(G) \). We initiate the study of the double Italian domatic number, and we present different sharp bounds on \( dd_I(G) \). In addition, we determine the double Italian domatic number of some classes of graphs.

We define the push statistic on permutations and multipermutations and use this to obtain various results measuring the degree to which an arbitrary permutation deviates from sorted order. We study the distribution on permutations for the statistic recording the length of the longest push and derive an explicit expression for its first moment and generating function. Several auxiliary concepts are also investigated. These include the number of cells that are not pushed; the number of cells that coincide before and after pushing (i.e., the fixed cells of a permutation); and finally the number of groups of adjacent columns of the same height that must be reordered at some point during the pushing process.

Let \( \mathcal{K} \) be a family of sets in \( \mathbb{R}^d \). For each countable subfamily \( \{K_m : m \geq 1\} \) of \( \mathcal{K} \), assume that \( \bigcap \{K_m : m \geq 1\} \) is consistent relative to staircase paths and starshaped via staircase paths, with a staircase kernel that contains a convex set of dimension \( d \). Then \( \bigcap \{K : K \in \mathcal{K} \} \) has these properties as well. For \( n \) fixed, \( n \geq 1 \), an analogous result holds for sets starshaped via staircase \( n \)-paths.