We show that the square of every connected S(K1,4)-free graph satisfying a matching condition has 2-connected spanning subgraph maximum degree at most 3. Furthermore, we characterise trees whose k. This generalises results on S(K1,3)-free graphs Henry and Vogler [7] Harary Schwenk [6], respectively.

We defined the symmetric TSP in lecture 6. The best known approximation algorithm for STSP is the 3/2 approximation algorithm of Christofides. The algorithm is very easy to describe, first choose a minimum cost spanning tree and then add that minimum cost Eulerian augmentation to the tree, that is the minimum cost set of edges to make the tree Eulerian. An undirected graph is Eulerian when ever...

A novel and effective approach is presented in this paper for solving the wellknown difficult problem of subgraph isomorphism. By integrating planning strategies with the concept of centered spanning tree, the presented approach achieves an average-case, near-linear performance. A majority of practical subgraph isomorphism problems deals with the cases of labeled graphs, which contains the NPco...

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. Our main result implies that, given any optimal colouring a sufficiently large complete K2n, there exists decomposition K2n into isomorphic spanning trees. This settles conjectures Brualdi–Hollingsworth (from 1996) and Constantine 2002) for graphs.

18] V. Ramachandran. Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. A linear-time algorithm for a special case of disjoint set union. Transitive com-paction in parallel via branchings. 47 sequence L(u) (see preprocessing stage) of endpoints of edges of C in T H and the preorder numbers of the children of u in T H , this can be done in time O(jCj +...

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...

We consider a special subgraph of a weighted directed graph: one comprising only the k heaviest edges incoming to each vertex. We show that the maximum weight branching in this subgraph closely approximates the maximum weight branching in the original graph. Specifically, it is within a factor of k k+1 . Our interest in finding branchings in this subgraph is motivated by a data compression appl...

An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the identifying code number (the size of a smallest identifying code), which indeed is not even a monotone parameter with respect to graph inclusion. We show that every...

We consider the existence of several different kinds of factors in 4-connected claw-free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4-connected line graph is hamiltonian, i.e. has a connected 2-factor. Conjecture 2 (Matthews and Sumner): Every 4-connected claw-free graph is hamil...

We study an NP-hard problem motivated by energy-efficiently maintaining the connectivity of a symmetric wireless sensor communication network. Given an edge-weighted n-vertex graph, find a connected spanning subgraph of minimum cost, where the cost is determined by letting each vertex pay the most expensive edge incident to it in the subgraph. We provide an algorithm that works in polynomial ti...