Kayles is the game, where two players alternately choose a vertex that has not been chosen before nor is adjacent to an already chosen vertex from a given graph. The last player that choses a vertex wins the game. We show, with help of Sprague-Grundy theory, that the problem to determine which player has a winning strategy for a given graph, can be solved in O(n3) time on interval graphs, on ci...

a set $s$ of vertices in a graph $g=(v,e)$ is called a total$k$-distance dominating set if every vertex in $v$ is withindistance $k$ of a vertex in $s$. a graph $g$ is total $k$-distancedomination-critical if $gamma_{t}^{k} (g - x) < gamma_{t}^{k}(g)$ for any vertex $xin v(g)$. in this paper,we investigate some results on total $k$-distance domination-critical of graphs.

The eccentricity of a vertex $v$ is the maximum distance between $v$ and anyother vertex. A vertex with maximum eccentricity is called a peripheral vertex.The peripheral Wiener index $ PW(G)$ of a graph $G$ is defined as the sum ofthe distances between all pairs of peripheral vertices of $G.$ In this paper, weinitiate the study of the peripheral Wiener index and we investigate its basicproperti...

Superatomic Molecules In article number 2300024, Yuichi Negishi and co-workers present a review focusing on vertex-shared linear M12n+1 superatomic molecules (M = noble metal), which discusses the most suitable ligand combinations for their synthesis, along with valence electron numbers, stability, optical absorption characteristics, luminescence properties of molecules.

The edge version of Szeged index and vertex version of PI index are defined very recently. They are similar to edge-PI and vertex-Szeged indices, respectively. The different versions of Szeged and PIindices are the most important topological indices defined in Chemistry. In this paper, we compute the edge-Szeged and vertex-PIindices of some important classes of benzenoid systems.

A series of genome-scale algorithms and high-performance implementations is described and shown to be useful in the genetic analysis of gene transcription. With them it is possible to address common questions such as: “are the sets of genes coexpressed under one type of conditions the same as those sets co-expressed under another?” A new noise-adaptive graph algorithm, dubbed “paraclique,” is i...

An on-line vertex coloring algorithm receives vertices of a graph in some externally determined order. Each new vertex is presented together with a set of the edges connecting it to the previously presented vertices. As a vertex is presented, the algorithm assigns it a color which cannot be changed afterwards. The on-line coloring problem was addressed for many different classes of graphs defin...

Gerke et al. (2019) introduced a game-theoretic model to study public good provision in social networks when there are constraints on sharing. This generates purely graph-theoretic problem termed exact capacitated domination. In the we given graph, graph with parameter defined each vertex that is interpreted as capacity of vertex. The objective find DP-Nash subgraph: spanning bipartite subgraph...

The vertex-edge Wiener index of a simple connected graph G is defined as the sum of distances between vertices and edges of G. Two possible distances D_1(u,e|G) and D_2(u,e|G) between a vertex u and an edge e of G were considered in the literature and according to them, the corresponding vertex-edge Wiener indices W_{ve_1}(G) and W_{ve_2}(G) were introduced. In this paper, we present exact form...