For the Firefighter Process with weights on the vertices, we show that the problem of deciding whether a subset of vertices of a total weight can be saved from burning remains NP-complete when restricted to binary trees. In addition, we show that a greedy algorithm that defends the vertex of highest degree adjacent to a burning vertex is not an \(\epsilon\)-\emph{approximation} algorithm for any \(\epsilon \in (0, 1]\) for the problem of determining the maximum weight that can be saved. This closes an open problem posed by MacGillivray and Wang.

Let \( K_v \) be the complete graph with \( v \) vertices. Let \( G \) be a finite simple graph. A \( G \)-decomposition of \( K_v \), denoted by \((v, G, 1)\)-GD, is a pair \((X, \mathcal{B})\), where \( X \) is the vertex set of \( K_v \), and \(\mathcal{B}\) is a collection of subgraphs of \( K_v \), called blocks, such that each block is isomorphic to \( G \). In this paper, the discussed graphs are \( G_i \), \( i = 1, 2, 3, 4 \), where \( G_i \) are four kinds of graphs with eight vertices and eight edges. We obtain the existence spectrum of \((v, G_i, 1)\)-GD.

We present a simple bijection between the set of triangulations of a convex polygon and the set of \(312\)-avoiding permutations.

A \({2-rainbow\; dominating\; function}\) of a graph \( G \) is a function \( g \) that assigns to each vertex a set of colors chosen from the set \( \{1, 2\} \) so that for each vertex \( v \) with \( g(v) = \emptyset \) we have \( \cup_{u \in N(v)} g(u) = \{1, 2\} \). The minimum of \( g(V(G)) = \sum_{v \in V(G)} |g(v)| \) over all such functions is called the \({2-rainbow \;domination\; number}\) \( \gamma_{r2}(G) \). A \(2\)-rainbow dominating function \( g \) of a graph \( G \) is independent if no two vertices assigned non-empty sets are adjacent. The \({independent \;2-rainbow\; domination\; number}\) \( i_{r2}(G) \) is the minimum weight of an independent \(2\)-rainbow dominating function of \( G \). In this paper, we study independent \(2\)-rainbow domination in graphs. We present some bounds and relations with other domination parameters.

For a connected graph \( G \) of order at least \( 3 \) and an integer \( k \geq 2 \), a \({twin\; edge}\) \( k \)-coloring of \( G \) is a proper edge coloring of \( G \) with the elements of \( \mathbb{Z}_k \), so that the induced vertex coloring in which the color of a vertex \( v \) in \( G \) is the sum (in \( \mathbb{Z}_k \)) of the colors of the edges incident with \( v \) is a proper vertex coloring. The minimum \( k \) for which \( G \) has a twin edge \( k \)-coloring is called the \({twin \;chromatic\; index}\) of \( G \) and is denoted by \( \chi_t'(G) \). It was conjectured that \( \Delta(T) \leq \chi_t'(T) \leq 2 + \Delta(T) \) for every tree of order at least \( 3 \), where \( \Delta(T) \) is the maximum degree of \( T \). This conjecture is verified for several classes of trees, namely brooms, double stars, and regular trees.

For a nontrivial connected graph \( G \), let \( c: V(G) \to \mathbb{Z}_2 \) be a vertex coloring of \( G \) where \( c(v) \neq 0 \) for at least one vertex \( v \) of \( G \). Then the coloring \( c \) induces a new coloring \( \sigma: V(G) \to \mathbb{Z}_2 \) of \( G \) defined by
\[
\sigma(v) = \sum_{u \in N[v]} c(u)
\]
where \( N[v] \) is the closed neighborhood of \( v \) and addition is performed in \( \mathbb{Z}_2 \). If \( \sigma(v) = 0 \in \mathbb{Z}_2 \) for every vertex \( v \) in \( G \), then the coloring \( c \) is called a (modular) monochromatic \( (2,0) \)-coloring of \( G \). A graph \( G \) having a monochromatic \( (2,0) \)-coloring is a (monochromatic) \( (2,0) \)-colorable graph. The minimum number of vertices colored \( 1 \) in a monochromatic \( (2,0) \)-coloring of \( G \) is the \( (2,0) \)-chromatic number of \( G \) and is denoted by \( \chi_{(2,0)}(G) \). For a \( (2,0) \)-colorable graph \( G \), the monochromatic \( (2,0) \)-spectrum \( S_{(2,0)}(G) \) of \( G \) is the set of all positive integers \( k \) for which exactly \( k \) vertices of \( G \) can be colored \( 1 \) in a monochromatic \( (2,0) \)-coloring of \( G \). Monochromatic \( (2,0) \)-spectra are determined for several well-known classes of graphs. If \( G \) is a connected graph of order \( n \geq 2 \) and \( a \in S_{(2,0)}(G) \), then \( a \) is even and \( 1 \leq |S_{(2,0)}(G)| \leq \left\lfloor \frac{n}{2} \right\rfloor \). It is shown that for every pair \( k,n \) of integers with \( 1 \leq k \leq \left\lfloor \frac{n}{2} \right\rfloor \), there is a connected graph \( G \) of order \( n \) such that \( |S_{(2,0)}(G)| = k \). A set \( S \) of positive even integers is \( (2,0) \)-realizable if \( S \) is the monochromatic \( (2,0) \)-spectrum of some connected graph. Although there are infinitely many non-\((2,0)\)-realizable sets, it is shown that every set of positive even integers is a subset of some \( (2,0) \)-realizable set. Other results and questions are also presented on \( (2,0) \)-realizable sets in graphs.

For two graphs \( H \) and \( G \), a decomposition \( \mathcal{D} = \{H_1, H_2, \ldots, H_k, R\} \) of \( G \) is called an \( H \)-maximal \( k \)-decomposition if \( H_i \cong H \) for \( 1 \leq i \leq k \) and \( R \) contains no subgraph isomorphic to \( H \). Let \(\text{Min}(G, H)\) and \(\text{Max}(G, H)\) be the minimum and maximum \( k \), respectively, for which \( G \) has an \( H \)-maximal \( k \)-decomposition. A graph \( G \) without isolated vertices is said to possess the intermediate decomposition property if for each connected graph \( G \) and each integer \( k \) with \(\text{Min}(G, H) \leq k \leq \text{Max}(G, H)\), there exists an \( H \)-maximal \( k \)-decomposition of \( G \). For a set \( S \) of graphs and a graph \( G \), a decomposition \( \mathcal{D} = \{H_1, H_2, \ldots, H_k, R\} \) of \( G \) is called an \( S \)-maximal \( k \)-decomposition if \( H_i \cong H \) for some \( H \in S \) for each integer \( i \) with \( 1 \leq i \leq k \) and \( R \) contains no subgraph isomorphic to any subgraph in \( S \). Let \(\text{Min}(G, S)\) and \(\text{Max}(G, S)\) be the minimum and maximum \( k \), respectively, for which \( G \) has an \( S \)-maximal \( k \)-decomposition. A set \( S \) of graphs without isolated vertices is said to possess the intermediate decomposition property if for every connected graph \( G \) and each integer \( k \) with \(\text{Min}(G, S) \leq k \leq \text{Max}(G, S)\), there exists an \( S \)-maximal \( k \)-decomposition of \( G \). While all those graphs of size \( 3 \) have been determined that possess the intermediate decomposition property, as have all sets consisting of two such graphs, here all remaining sets of graphs having size \( 3 \) that possess the intermediate decomposition property are determined.

An Eulerian graph \( G \) of size \( m \) is said to satisfy the Eulerian Cycle Decomposition Conjecture if the minimum number of odd cycles in a cycle decomposition of \( G \) is \( a \), the maximum number of odd cycles in a cycle decomposition is \( b \), and \( \ell \) is an integer such that \( a \leq \ell \leq b \) where \( \ell \) and \( m \) are of the same parity, then there is a cycle decomposition of \( G \) with exactly \( \ell \) odd cycles. Several regular complete \( 5 \)-partite graphs are shown to have this property.

An \( H_3 \) graph is a multigraph on three vertices with double edges between two pairs of distinct vertices and a single edge between the third pair. To settle the \( H_3 \) decomposition problem completely, one needs to complete the decomposition of a \( 2K_{10t+5} \) into \( H_3 \) graphs. In this paper, we present two new construction methods for such decompositions, resulting in previously unknown decompositions for \( v = 15, 25, 35, 45 \) and two new infinite families.

Analog modulation has served us very well over the years. Digital modulation is an improvement over analog modulation because it provides better bandwidth utilization over analog modulation, less power for signal propagation, it is natural for packet transmission, forward error correction, automatic repeat request, encryption, compression, and signal transformation so that it looks like noise to the adversary. Digital wireless communication is an enormous area that is rapidly growing. Digital communication is a field in which theoretical ideas have had an unusually powerful impact on system design and practice. In this research paper we provide a digital modulation algorithm for efficient transmission based on circular probability distribution theory.