For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the complete graph on n vertices, which Maker wins as soon as the graph she builds contains a copy of T . We prove that if T has bounded maximum degree, then Maker can win this game within n+ 1 moves. Moreover, we prove that Maker can build almost every tree on n vertices in n−1 moves and provide non-trivial ...

We investigate the smallest number λ(G) of vertices that need to be removed from a non-empty graph G so that the resulting graph has a smaller maximum degree. We prove that if n is the number of vertices, k is the maximum degree, and t is the number of vertices of degree k, then λ(G) ≤ n+(k−1)t 2k . We also show that λ(G) ≤ n k+1 if G is a tree. These bounds are sharp. We provide other bounds t...

By a theorem of Mader [5], highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. Solving a problem of Diestel [2], we extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex-degree (or multiplicity) for the ends of the graph, i.e. a large number of disjoint rays in each end. We give...

One of the biggest huddles faced by researchers studying algorithms for massive graphs is the lack of large input graphs that are essential for the development and test of the graph algorithms. This paper proposes two efficient and highly scalable parallel graph generation algorithms that can produce massive realistic graphs to address this issue. The algorithms, designed to achieve high degree...

A vertex dominating path in a graph is a path P such that every vertex outside P has a neighbor on P . In 1988 H. Broersma stated a result implying that every n-vertex kconnected graph G such that σ(k+2)(G) ≥ n− 2k − 1 contains a dominating path. We show that every n-vertex k-connected graph with σ2(G) ≥ 2n k+2 + f(k) contains a dominating path of length at most O(|T |), where T is a minimum do...

the harmonic index h(g) , of a graph g is defined as the sum of weights 2/(deg(u)+deg(v)) of all edges in e(g), where deg (u) denotes the degree of a vertex u in v(g). in this paper we define the harmonic polynomial of g. we present explicit formula for the values of harmonic polynomial for several families of specific graphs and we find the lower and upper bound for harmonic index in caterpill...

For a graph $G$ with edge set $E(G)$, the multiplicative sum Zagreb index of $G$ is defined as$Pi^*(G)=Pi_{uvin E(G)}[d_G(u)+d_G(v)]$, where $d_G(v)$ is the degree of vertex $v$ in $G$.In this paper, we first introduce some graph transformations that decreasethis index. In application, we identify the fourteen class of trees, with the first through fourteenth smallest multiplicative sum Zagreb ...

A triangulation is said to be even if each vertex has even degree. For even triangulations, define the N -flip and the P2-flip as two deformations preserving the number of vertices. We shall prove that any two even triangulations on the sphere with the same number of vertices can be transformed into each other by a sequence of N and P2-flips.

We show that, for all choices of integers k > 2 and m, there are simple 3connected k-crossing-critical graphs containing more than m vertices of each even degree ≤ 2k − 2. This construction answers one half of a question raised by Bokal, while the other half asking analogously about vertices of odd degrees at least 7 in crossing-critical graphs remains open. Furthermore, our newly constructed g...

An extended star is a tree which has only one vertex with degree larger than two. The -center problem in a graph  asks to find a subset  of the vertices of  of cardinality  such that the maximum weighted distances from  to all vertices is minimized. In this paper we consider the -center problem on the unweighted extended stars, and present some properties to find solution.