Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the results of adding an inclusion minimal set of edges to produce a triangulation. This topic was rst studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we k...

Lexicographic Breadth First Search (LBFS) is one of fundamental graph search algorithms that has numerous applications, including recognition of graph classes, computation of graph parameters, and detection of certain graph structures. The well-known result on the end-vertices of LBFS of chordal graphs has tempted researchers to study the end-vertices of LBFS of various classes of graphs, inclu...

A locally connected spanning tree (LCST) T of a graph G is a spanning tree of G such that for each node its neighborhood in T induces a connected subgraph in G. The problem of determining whether a graph contains an LCST or not has been proved to be NP-complete, even if the graph is planar or chordal. The main result of this paper is a linear time algorithm that, given an SC 3-tree (i.e. a maxi...

We extend Crapo’s β invariant from matroids to greedoids, concentrating especially on antimatroids. Several familiar expansions for β(G) have greedoid analogs. We give combinatorial interpretations for β(G) for simplicial shelling antimatroids associated with chordal graphs. When G is this antimatroid and b(G) is the number of blocks of the chordal graph G, we prove β(G) = 1− b(G).

In this paper we introduce a new notion of collective tree spanners. We say that a graph G = (V,E) admits a system of μ collective additive tree r-spanners if there is a system T (G) of at most μ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT (x, y) ≤ dG(x, y) + r. Among other results, we show that any chordal graph, chordal bipartite ...

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bound...

We show that the independence complex of a chordal graph is contractible if and only if this complex is dismantlable (strong collapsible) and it is homotopy equivalent to a sphere if and only if its core is a cross-polytopal sphere. The proof uses the properties of tree models of chordal graphs.

The clique cover width of G, denoted by ccw(G), is the minimum value of the bandwidth of all graphs that are obtained by contracting the cliques in a clique cover of G into a single vertex. For i = 1, 2, ..., d, let Gi be a graph with V (Gi) = V , and let G be a graph with V (G) = V and E(G) = ∩di=1(Gi), then we write G = ∩di=1Gi and call each Gi, i = 1, 2, ..., d a factor of G. The case where ...

Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we ...