On graphs whose maximal subgraphs have at most two orbits

Two-connected Spanning Subgraphs with at Most

On Graphs whose Spread is Maximal

A graph’s spread is defined as the difference between the largest eigenvalue and the least eigenvalue of the graph’s adjacency matrix. Characterizing a graph with maximal spread is still a difficult problem. If we restrict the discussion to some classes of connected graphs of a prescribed order and size, it simplifies the problem and it may allow us to solve it. Here, we discuss some results on...

Two minimal forbidden subgraphs for double competition graphs of posets of dimension at most two

Let S be any set of points in the Euclidean plane R2. For any p = (x, y) ∈ S, put SW (p) = {(x, y) ∈ S : x < x and y < y} and NE(p) = {(x, y) ∈ S : x > x and y > y}. Let GS be the graph with vertex set S and edge set {pq : NE(p) ∩ NE(q) 6= ∅ and SW (p) ∩ SW (q) 6= ∅}. We prove that the graphH with V (H) = {u, v, z, w, p, p1, p2, p3} and E(H) = {uv, vz, zw, wu, p1p3, p2p3, pu, pv, pz, pw, pp1, p...

Spanning trees whose stems have at most k leaves

Let T be a tree. A vertex of T with degree one is called a leaf, and the set of leaves of T is denoted by Leaf(T ). The subtree T − Leaf(T ) of T is called the stem of T and denoted by Stem(T ). A spanning tree with specified stem was first considered in [3]. A tree whose maximum degree at most k is called a k-tree. Similarly, a stem whose maximum degree at most k in it is called a k-stem, and ...

On dominating sets whose induced subgraphs have a bounded diameter

We study dominating sets whose induced subgraphs have a bounded diameter. Such sets were used in recent papers by Kim et al. and Yu et al. to model virtual backbones in wireless networks where the number of hops required to forward messages within the backbone is minimized. We prove that for any fixed k ≥ 1 it is NP-complete to decide whether a given graph admits a dominating set whose induced ...