This paper describes an extremely fast polynomial time algorithm, the NOVCA (Near Optimal Vertex Cover Algorithm) that produces an optimal or near optimal vertex cover for any known undirected graph G (V, E). NOVCA is based on the idea of (1) including the vertex having maximum degree in the vertex cover and (2) rendering the degree of a vertex to zero by including all its adjacent vertices. Th...

We show that the Min Cut Linear Arrangement Problem (Min Cut) is NP-complete for trees with polynomial size edge weights and derive from this the NP-completeness of Min Cut for planar graphs with maximum vertex degree 3. This is used to show the NP-completeness of Search Number, Vertex Separation, Progressive Black/ White Pebble Demand, and Topological Bandwidth for planar graphs with maximum v...

A vertex v of a graph G is a boundary vertex if there exists a vertex u such that the distance in G from u to v is at least the distance from u to any neighbour of v. We give the best possible lower bound, up to a constant factor, on the number of boundary vertices of a graph in terms of its minimum degree (or maximum degree). This settles a problem introduced by Hasegawa and Saito.

A graph $G$ is $d$-degenerate if every non-null subgraph of has a vertex degree at most $d$. We prove that $n$-vertex planar $3$-degenerate induced order least $3n/4$.

In shadow volume rendering, the shadow volume silhouette edges are used to create primitives that model the shadow volume. A common misconception is that the vertices on such silhouettes can only be connected to two silhouette edges, i.e., have degree two. Furthermore, some believe that the degree of such a vertex can have any degree. In this short note, we present a geometric proof that shows ...

We consider the minimum feedback vertex set problem for some bipartite graphs and degree-constrained graphs. We show that the problem is linear time solvable for bipartite permutation graphs and NP-hard for grid intersection graphs. We also show that the problem is solvable in O(n2 log6 n) time for n-vertex graphs with maximum degree at most three. key words: 3-regular graph, bipartite permutat...

The vertex-connectivity and edge-connectivity of the zero-divisor graph associated to a finite commutative ring are studied. It is shown that the edgeconnectivity of ΓR always coincides with the minimum degree. When R is not local, it is shown that the vertex-connectivity also equals the minimum degree, and when R is local, various upper and lower bounds are given for the vertex-connectivity.

This work investigates the parameterized complexity of three related graph modification problems. Given a directed graph, a distinguished vertex, and a positive integer k, Minimum Indegree Deletion asks for a vertex subset of size at most k whose removal makes the distinguished vertex the only vertex with minimum indegree. Minimum Degree Deletion is analogously defined, but deals with undirecte...

Corrádi and Hajnal [1] showed that any graph of order at least 3k with minimum degree at least 2k contains k vertex-disjoint cycles. This minimum degree condition is sharp, because the complete bipartite graph K2k−1,n−2k+1 does not contain k vertex-disjoint cycles. About the existence of vertex-disjoint cycles of the same length, Thomassen [4] conjectured that the same minimum degree condition ...

Corradi and Hajnal proved that a graph of order at least 3k and minimum degree at least 2k contains k vertex-disjoint cycles. Häggkvist subsequently conjectured that a sufficiently large graph of minimum degree at least four contains two vertex-disjoint cycles of the same length. We prove that this conjecture is correct. In doing so, we give a short proof of the known result that if k > 2, ther...