A Krausz partition of a graph G is a partition of the edges of G into complete subgraphs. The Krausz dimension of a graph G is the least number k such that G admits a Krausz partition in which each vertex belongs to at most k classes. The graphs with Krausz dimension at most 2 are exactly the line graphs, and graphs of the Krausz dimension at most k are intersection graphs of k-uniform linear h...

An incidence of G consists of a vertex and one of its incident edge in G. The incidence coloring problem is a variation of vertex coloring problem. The problem is to find the minimum number (called incidence coloring number) of colors assigned to every incidence of G so that the adjacent incidences are not assigned the same color. In this paper, we propose a linear time algorithm for incidence-...

Let G be a simple undirected graph on n vertices, and let I(G) ⊆ R = k[x 1 ,. .. , x n ] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and He...

Chordal graphs can be used to encode dependency models that are representable by both directed acyclic and undirected graphs. This paper discusses a very simple and efficient algorithm to learn the chordal structure of a probabilistic model from data. The algorithm is a greedy hillclimbing search algorithm that uses the inclusion boundary neighborhood over chordal graphs. In the limit of a larg...

We show that the class of chordal claw-free graphs admits LREC=-definable canonization. LREC= is a logic that extends first-order logic with counting by an operator that allows it to formalize a limited form of recursion. This operator can be evaluated in logarithmic space. It follows that there exists a logarithmic-space canonization algorithm, and therefore a logarithmic-space isomorphism tes...

A set K of nodes of a graph G is geodesically convex (respectively, monophonically convex) if K contains every node on every shortest (respectively, chordless) path joining nodes in K. We investigate the classes of graphs which are characterized by certain local convexity conditions with respect to geodesic convexity, in particular, those graphs in which balls around nodes are convex, and those...

Problem statement: Modified Chordal Rings Degree Four, called CHRm4 is the first modified structure of chordal rings. This CHRm4 is an undirected circulant graph and is a double loop graph. Approach: This study presented the main properties of CHRm4. There are connectivity, Hamiltonian cycle and asymmetric. Results: Several definitions, postulates, corollary, theorems and lemmas were constructe...

Motivated by the conjecture on the L(2, 1)-labelling number λ(G) of a graph G by Griggs and Yeh [2] and the question: “Is the upper bound (∆+3)/4 for λ(G) for chordal graphs with maximum degree ∆ is sharp?”, posed by Sakai [3], we study the bounds for λ(G) for chordal graphs in this paper. Let G be a chordal graph on n vertices with maximum degree ∆ and maximum clique number ω. We improve the u...

A set D ⊆ V of a graph G = (V,E) is called a restrained dominating set of G if every vertex not in D is adjacent to a vertex in D and to a vertex in V \D. The MINIMUM RESTRAINED DOMINATION problem is to find a restrained dominating set of minimum cardinality. Given a graph G, and a positive integer k, the RESTRAINED DOMINATION DECISION problem is to decide whether G has a restrained dominating ...

A partitioning problem on chordal graphs that arises in the solution of sparse triangular systems of equations on parallel computers is considered Roughly the problem is to partition a chordal graph G into the fewest transitively orientable subgraphs over all perfect elimination orderings of G subject to a certain precedence relationship on its vertices In earlier work a greedy scheme that solv...