Acyclic homomorphisms to stars of graph Cartesian products and chordal bipartite graphs

AbstractHomomorphisms to a given graph H (H-colourings) are considered in the literature among other graph colouring concepts. We restrict our attention to a special class of H-colourings, namely H is assumed to be a star. Our additional requirement is that the set of vertices of a graph G mapped into the central vertex of the star and any other colour class induce in G an acyclic subgraph. We investigate the existence of such a homomorphism to a star of given order. The complexity of this problem is studied. Moreover, the smallest order of a star for which a homomorphism of a given graph G with desired features exists is considered. Some exact values and many bounds of this number for chordal bipartite graphs, cylinders, grids, in particular hypercubes, are given. As an application of these results, we obtain some bounds on the cardinality of the minimum feedback vertex set for specified graph classes

Feedback vertex number of Sierpi\'{n}ski-type graphs

The feedback vertex number $\tau(G)$ of a graph $G$ is the minimum number of vertices that can be deleted from $G$ such that the resultant graph does not contain a cycle. We show that $\tau(S_p^n)=p^{n-1}(p-2)$ for the Sierpi\'{n}ski graph $S_p^n$ with $p\geq 2$ and $n\geq 1$. The generalized Sierpi\'{n}ski triangle graph $\hat{S_p^n}$ is obtained by contracting all non-clique edges from the Sierpi\'{n}ski graph $S_p^{n+1}$. We prove that $\tau(\hat{S}_3^n)=\frac {3^n+1} 2=\frac{|V(\hat{S}_3^n)|} 3$, and give an upper bound for $\tau(\hat{S}_p^n)$ for the case when $p\geq 4$

Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

Graphs and Algorithm

Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

Graphs and Algorithm