The design whose blocks consist of all \(k\)-element multisets drawn from a \(v\)-set, denoted \(M(v,k)\), is a classical example of a balanced \((k+1)\)-ary design. Although its parameters are well known, existing derivations often rely on general multiset design theory. This paper gives unified elementary derivations of the parameters \(b\), \(r\), and \(\lambda\) using stars-and-bars and double counting. We exhibit a natural multiplicity-layer decomposition: removing \(s\) copies of a fixed point from all blocks in which it has multiplicity exactly \(s\) yields a family of subdesigns naturally in bijection with \(M(v-1,k-s)\). This viewpoint clarifies the recursive structure underlying complete multiset designs. Finally, the multiplicity vectors of blocks of \(M(v,k)\) form a \((k+1)\)-ary code of length \(v\) with constant coordinate sum \(k\) and minimum Hamming distance \(2\), achieving size \(\binom{v+k-1}{k}\).

The Explorer-Director game, first introduced by Nedev and Muthukrishnan (2008), simulates a Mobile Agent exploring a ring network with an inconsistent global sense of direction. Two players, the Explorer and the Director, jointly control a token’s movement on the vertices of a graph G with initial location v. Each turn, the Explorer calls any valid distance, d, aiming to maximize the number of vertices the token visits, and the Director moves the token to any vertex distance d away aiming to minimize the number of visited vertices. The game ends when no new vertices can be visited, assuming optimal play, and we denote the total number of visited vertices by f_d(G, v). Here we study a variant where, if the token is on vertex u, the Explorer is allowed to select any valid path length, ℓ, and the Director now moves the token to any vertex v such that G contains a uv path of length ℓ. The corresponding parameter is f_p(G, v). In this paper, we explore how far apart f_d(G, v) and f_p(G, v) can be, proving that for any n there are graphs G and H with f_p(G, v) − f_d(G, v) > n and f_d(H, v) − f_p(H, v) > n.

We study edge partitions of a bipartite graph into induced-2K₂-free bipartite graphs, i.e. into Ferrers (chain) graphs. We define fp(G) as the minimum number of parts in such a partition. We prove general lower and upper bounds in terms of induced matchings and Dilworth widths of neighborhood posets. We compute the parameter exactly for paths and even cycles, and we exhibit separations showing that the induced-matching lower bound and the width upper bound can both be far from tight. We also record a simple host-induced conflict-graph lower bound, present a 0–1 matrix viewpoint, and add some complexity remarks.

We study the Equivalent Local Sequence Problem (ELSP), which consists in recovering an explicit sequence of local complementations that transforms a graph into a locally equivalent one. Focusing on directed Paley graphs, we establish that local complementations commute and induce a free action of an elementary abelian 2-group. The stabilizer condition is reformulated as a system of convolution equations and analyzed through Fourier techniques over finite fields, leading to a proof of stabilizer triviality. As a consequence, each graph in the orbit admits a unique subset encoding, and ELSP reduces to solving a linear inversion problem over 𝔽₂. This characterization completely resolves ELSP for directed Paley graphs, provides a polynomial-time inversion algorithm and highlights structural features that may support future developments in cryptographic frameworks and quantum graph-state models.

Given a configuration of pebbles on the edges of a connected graph G, an edge pebbling move is defined as the removal of two pebbles off an edge and placing one on an adjacent edge. The domination cover edge pebbling number of a graph G is the minimum number of pebbles required such that the set of edges that contain pebbles form an edge dominating set S of G, for the initial configuration of pebbles can be altered by a sequence of pebbling moves and it is denoted by ψ_e(G) for a graph G. In this paper, we determine ψ_e(G) for Generalized Petersen graph, Jewel graph and Triangular snake graph.