20 research outputs found
An algorithm for finding homogeneous pairs
AbstractA homogeneous pair in a graph G = (V, E) is a pair Q1, Q2 of disjoint sets of vertices in this graph such that every vertex of V (Q1 ∪ Q2) is adjacent either to all vertices of Q1 or to none of the vertices of Q1 and is adjacent either to all vertices of Q2 or to none of the vertices of Q2. Also ¦Q1¦ ⩾ 2 or ¦Q2¦⩾ 2 and ¦V (Q1 ∪ Q2)¦ ⩾ 2. In this paper we present an O(mn3)-time algorithm which determines whether a graph contains a homogeneous pair, and if possible finds one
Some Structural Resuits on Prime Graphs
Given a graph G = (V,E), a subset M of V is a module [17] (or an interval [10] or an autonomous [11] or a clan [8] or a homogeneous set [7] ) of G provided that x ∼ M for each vertex x outside M. So V,φ and {x}, where x ∈ V , are modules of G, called trivial modules. The graph G is indecomposable [16] if all the modules of G are trivial. Otherwise we say that G is decomposable . The prime graph G is an indecomposable graph with at least four vertices. Let G and H be two graphs. Let If G has no induced subgraph isomorphic to H, then we say that G is H-free. In this paper, we will prove the next theore
Convex circuit free coloration of an oriented graph
We introduce the \textit{convex circuit-free coloration} and \textit{convex circuit-free chromatic number} of an oriented graph and establish various basic results. We show that the problem of deciding if an oriented graph verifies is NP-complete if and polynomial if . We exhibit an algorithm which finds the optimal convex circuit-free coloration for tournaments, and characterize the tournaments that are \textit{vertex-critical} for the convex circuit-free coloration
3-uniform hypergraphs: modular decomposition and realization by tournaments
Let be a 3-uniform hypergraph. A tournament defined on is
a realization of if the edges of are exactly the 3-element subsets of
that induce 3-cycles. We characterize the 3-uniform hypergraphs that
admit realizations by using a suitable modular decomposition
Deciding whether there are infinitely many prime graphs with forbidden induced subgraphs
A homogeneous set of a graph G is a set X of vertices such that 2≤|X|V(G)| and no vertex in V(G)−X has both a neighbor and a non-neighbor in X. A graph is prime if it has no homogeneous set. We present an algorithm to decide whether a class of graphs given by a finite set of forbidden induced subgraphs contains infinitely many non-isomorphic prime graphs
A representation for the modules of a graph and applications
We describe a simple representation for the modules of a graph C. We show that the modules of C are in one-to-one correspondence with the ideaIs of certain posets. These posets are characterizaded and shown to be layered posets, that is, transitive closures of bipartite tournaments. Additionaly, we describe applications of the representation. Employing the above correspondence, we present methods for solving the following problems: (i) generate alI modules of C, (ii) count the number of modules of C, (iii) find a maximal module satisfying some hereditary property of C and (iv) find a connected non-trivial module of C
Graphs whose indecomposability graph is 2-covered
Given a graph , a subset of is an interval of provided
that for any and , if and only
if . For example, , and are
intervals of , called trivial intervals. A graph whose intervals are trivial
is indecomposable; otherwise, it is decomposable. According to Ille, the
indecomposability graph of an undirected indecomposable graph is the graph
whose vertices are those of and edges are the unordered
pairs of distinct vertices such that the induced subgraph is indecomposable. We characterize the indecomposable
graphs whose admits a vertex cover of size 2.Comment: 31 pages, 5 figure
On prime Cayley graphs
The decomposition of complex networks into smaller, interconnected components
is a central challenge in network theory with a wide range of potential
applications. In this paper, we utilize tools from group theory and ring theory
to study this problem when the network is a Cayley graph. In particular, we
answer the following question: Which Cayley graphs are prime