In this paper, we look at the problem of randomized leader election in synchronous distributed networks with a special focus on message complexity. We provide an algorithm that solves implicit version (where non-leader nodes need not be aware identity leader) any general network $$O(\sqrt{n} \log ^{7/2} n \cdot t_{mix})$$ messages and $$O(t_{mix}\log ^2 n)$$ time, where is number $$t_{mix}$$ re...

The generalized bull is the graph B i , j ${B}_{i,j}$ obtained by attaching endvertices of two disjoint paths lengths $i,j$ to vertices a triangle. We prove that every 3-connected { K 1 3 X } $\{{K}_{1,3},X\}$ -free graph, where ? 6 2 5 4 $X\in \{{B}_{1,6},{B}_{2,5},{B}_{3,4}\}$ Hamilton-connected. results are sharp and complete characterization forbidden induced bulls implying Hamilton-connect...

In this paper, we introduce and initiate the study of outer-connected semitotal domination in graphs. Given a graph G without isolated vertices, set S vertices is dominating if every vertex outside adjacent to distance at most 2 units from another S. A an either = V (G) or ̸= satisfying property that subgraph induced by \ connected. The smallest cardinality ̃γt2(G) number ofG. First, determine sp...

The following result is proved. Every «-connected graph contains either a vertex whose removal results in a graph which is also «-connected or a vertex of degree less than (3n—1)/2. Introduction. A graph G is said to be n-connected if the removal of fewer than « vertices from G neither disconnects it nor reduces it to the trivial graph consisting of a single vertex. The maximum value of « for w...

We find the asymptotic number of connected graphs with k vertices and k − 1 + l edges when k, l approach infinity, reproving a result of Bender, Canfield and McKay. We use the probabilistic method, analyzing breadth-first search on the random graph G(k, p) for an appropriate edge probability p. Central is analysis of a random walk with fixed beginning and end which is tilted to the left.

A graph is connected-homogeneous if any isomorphism between finite connected induced subgraphs extends to an automorphism of the graph. In this paper we classify the countably infinite connectedhomogeneous graphs. We prove that if Γ is connected countably infinite and connected-homogeneous then Γ is isomorphic to one of: Lachlan and Woodrow’s ultrahomogeneous graphs; the generic bipartite graph...

Given a graph G and a positive integer k, denote by G[k] the graph obtained from G by replacing each vertex of G with an independent set of size k. A graph G is called pseudo-k Hamiltonian-connected if G[k] is Hamiltonian-connected, i.e., every two distinct vertices of G[k] are connected by a Hamiltonian path. A graph G is called pseudo Hamiltonian-connected if it is pseudo-k Hamiltonian-connec...

We prove that if G = (VG, EG) is a finite, simple, and undirected graph with κ components and independence number α(G), then there exist a positive integer k ∈ N and a function f : VG → N0 with non-negative integer values such that f(u) ≤ dG(u) for u ∈ VG, α(G) ≥ k ≥ ∑ u∈VG 1 dG(u)+1−f(u) , and ∑ u∈VG f(u) ≥ 2(k − κ). This result is a best-possible improvement of a result due to Harant and Schi...