The Firefighter problem: Saving sets of vertices on cubic graphs

The Firefighter problem was introduced by Hartnell in 1995 and corresponds to a scenario in which a fire breaks out at one or more vertices of a graph and spreads to all adjacent vertices that have not been protected in previous steps. In this paper, we present an algorithm of the Firefighter problem to an infinite family of full icosahedral symmetry fullerene graphs, providing the surviving rate at least 50%, contributing to the question proposed by Costa in 2015 on the surviving rate of graphs with maximum degree 3.

The firefighter problem: Further steps in understanding its complexity

In the vertex cover problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of vertices S of size at most k such that every edge of G is incident on at least one vertex in S. We study the vertex cover problem on graphs with maximum degree 4 and minimum degree at least 2, parameterized by $$r = k-n/3$$ . We give an algorithm for this problem whose running time is $$O^*(1.6253^r)$$ . As a corollary, we obtain an $$O^*(1.2403^k)$$ -time algorithm for vertex cover on graphs with maximum degree 4.

Extreme Wiener indices of trees with given number of vertices of maximum degree

Non-hamiltonian graphs in which every edge-contracted subgraph is hamiltonian

In this paper, we study lower bounds on the size of a maximum independent set and a maximum matching in a hypergraph of rank three. The rank in a hypergraph is the size of a maximum edge in the hypergraph. A hypergraph is simple if no two edges contain exactly the same vertices. Let H be a hypergraph and let and be the size of a maximum independent set and a maximum matching, respectively, in H, where a set of vertices in H is independent (also called strongly independent in the literature) if no two vertices in the set belong to a common edge. Let H be a hypergraph of rank at most three and maximum degree at most three. We show that with equality if and only if H is the Fano plane. In fact, we show that if H is connected and different from the Fano plane, then and we characterize the hypergraphs achieving equality in this bound. Using this result, we show that that if H is a simple connected 3‐uniform hypergraph of order at least 8 and with maximum degree at most three, then and there is a connected 3‐uniform hypergraph H of order 19 achieving this lower bound. Finally, we show that if H is a connected hypergraph of rank at most three that is not a complete hypergraph on vertices, where denotes the maximum degree in H, then and this bound is asymptotically best possible. © 2012 Wiley Periodicals, Inc. J. Graph Theory

We consider models of evolving networks $$\left\{ {\mathcal {G}}_n:n\ge 0\right\} $$ modulated by two parameters: an attachment function $$f:{\mathbb {N}}_0 \rightarrow {\mathbb {R}}_+$$ and a (possibly random) attachment sequence $$\left\{ m_i:i\ge 1\right\} $$ . Starting with a single vertex, at each discrete step $$i\ge 1$$ a new vertex $$v_i$$ enters the system with $$m_i\ge 1$$ edges which it sequentially connects to a pre-existing vertex $$v\in {\mathcal {G}}_{i-1}$$ with probability proportional to $$f(\text{ degree }(v))$$ . We consider the problem of emergence of persistent hubs: existence of a finite (a.s.) time $$n^*$$ such that for all $$n\ge n^*$$ the identity of the maximal degree vertex (or in general the K largest degree vertices for $$K\ge 1$$ ) does not change. We obtain general conditions on f and $$\left\{ m_i:i\ge 1\right\} $$ under which a persistent hub emerges, and also those under which a persistent hub fails to emerge. In the case of lack of persistence, for the specific case of trees ( $$m_i\equiv 1$$ for all i), we derive asymptotics for the maximal degree and the index of the maximal deg ree vertex (time at which the vertex with current maximal degree entered the system) to understand the movement of the maximal degree vertex as the network evolves. A key role in the analysis is played by an inverse rate weighted martingale constructed from a continuous time embedding of the discrete time model. Asymptotics for this martingale, including concentration inequalities and moderate deviations form the technical foundations for the main results.

The nilpotent graph is a graph’s structure motivated by the characteristics of some elements of a ring. If G is a graph, G(N) is a nilpotent graph in which the set of all non-nilpotent elements of a ring taken as a vertex set. For each x, y in R\ il(R), x and y are said to be connected or adjacent to an edge, if the sum of the vertices x and y is equal to the nilpotent element in the ring. This research aims to explain how to construct a nilpotent graph G(N) from a commutative ring. The results of this research are the characteristic of the nilpotent graph of a ring Zn for n prime numbers, a nilpotent graph for the ring Z 2 k, and ring Zn with n = 6m where 1 ≤ m < 5 can create 3 nilpotent subgraphs, they are a complete subgraph (km ) and 2 regular subgraphs in which the degree of vertices is equal to m.

Let be a simple connected graph with vertex set V and edge set E. The first, second and third Zagreb indices of G are defind, respectivly by: , and where is the degree of vertex u in G and uv is an edge of G, connecting the vertices u and v. Recently, the first and second multiplicative Zagreb indices of graph are defind by: and . The first and second Zagreb coindices of graph are defind by: and . and , named as multiplicative Zagreb coindices. In this article, we compute the first, second and the third Zagreb indices and the first and second multiplicative Zagreb indices of some graphs. The first and second Zagreb coindices and the first and second multiplicative Zagreb coindices of these graphs are also computed.

Moments in graphs

The degree distance of a connected graph G with vertex set V(G) is defined as D'(G)= ?u?V (G) dG (u)DG (u), where dG (u) denotes the degree of vertex u and DG (u) denotes the sum of distances between u and all vertices of G. We determine the maximum degree distance of n-vertex unicyclic graphs with given maximum degree, and the first seven maximum degree distances of n-vertex unicyclic graphs for n ? 6.

Let G be a simple graph with vertex set V(G) and edge set E(G). Automorphism of G is an isomorphism from G to itself. In other words, an automorphism of G is a permutation φ of V(G), which has the property that uv ∈ E(G) if and only if φ(u)φ(v) ∈ E(G), for every u, v ∈ V(G) that is φ preserves adjacency. The complete n-ary tree is a rooted-tree where its every internal vertex has exactly n children. The height of a rooted tree is the number of edges on longest path connecting the root to a leaf (a vertex of degree one). In this paper, we determine the number of automorphism of complete 2,3,4-ary trees with height h, for any positive integer h. We also conjecture the general formula to determine the number of automorphism of complete n-ary trees, for any n ≥ 5.

A bipartite graph G(X, Y) whose vertex set is partitioned into X and Y is a convex bipartite graph, if there is an ordering of $$X=(x_1,\ldots ,x_m)$$ such that for all $$y \in Y$$ , $$N_G(y)$$ is consecutive with respect to the ordering of X, and G is said to have convexity with respect to X. A k-star caterpillar is a tree with a collection of stars with each star having k vertices of degree one whose roots are joined by a path. For a bipartite graph with partitions X and Y, we associate a k-star caterpillar on X such that for each vertex in Y, its neighborhood induces a tree. The minimum Steiner tree problem (STREE) is defined as follows: given a connected graph $$G=(V,E)$$ and a subset of vertices $$R \subseteq V(G)$$ , the objective is to find a minimum cardinality set $$S \subseteq V(G)$$ such that the set $$R \cup S$$ induces a connected subgraph. In this paper, we present the following dichotomy result: we show that STREE is NP-complete for 1-star caterpillar convex bipartite graphs and polynomial-time solvable for 0-star caterpillar convex bipartite graphs (also known as convex bipartite graphs). We also strengthen the well-known result of Müller and Brandstädt (Theoret Comput Sci 53(2-3):257-265, 1987), which says STREE in chordal bipartite graphs is NP-complete (reduction instances are 3-star caterpillar convex bipartite graphs). As an application, we use our STREE results to solve: (i) the classical dominating set problem in convex bipartite graphs, (ii) STREE on interval graphs, linear time.

Ordered graphs and large bi-cliques in intersection graphs of curves

A graph is ‐colorable if its vertex set can be partitioned into sets , such that for each , the subgraph of induced by has maximum degree at most . The Four Color Theorem states that every planar graph is ‐colorable, and a classical result of Cowen, Cowen, and Woodall shows that every planar graph is ‐colorable. In this paper, we extend both of these results to graphs on surfaces. Namely, we show that every graph embeddable on a surface of Euler genus is ‐colorable and ‐colorable. Moreover, these graphs are also ‐colorable and ‐colorable. We also prove that every triangle‐free graph that is embeddable on a surface of Euler genus is ‐colorable. This is an extension of Grötzsch's Theorem, which states that triangle‐free planar graphs are ‐colorable. Finally, we prove that every graph of girth at least 7 that is embeddable on a surface of Euler genus is ‐colorable. All these results are best possible in several ways as the girth condition is sharp, the constant maximum degrees cannot be improved, and the bounds on the maximum degrees depending on are tight up to a constant multiplicative factor.