Indecomposable Graphs Research Articles

Prime Graph Generation through Single Edge Addition: Characterizing a Class of Graphs

A graph G consists of a finite set V (G) of vertices with a collection E(G) of unordered pairs of distinct vertices called edge set of G. Let G be a graph. A set M of vertices is a module of G if, for vertices x and y in M and each vertex z outside M, {z, x} ∈ E(G) ⇐⇒ {z, y} ∈ E(G). Thus, a module of G is a set M of vertices indistinguishable by the vertices outside M. The empty set, the singleton sets and the full set of vertices represent the trivial modules. A graph is indecomposable if all its modules are trivial, otherwise it is decomposable. Indecomposable graphs with at least four vertices are prime graphs. The introduction and the study of the construction of prime graphs obtained from a given decomposable graph by adding one edge constitue the central points of this paper.

Inverse limits of upper semicontinuous functions and indecomposable continua

We consider how inverse limits and Mahavier-products of upper semicontinuous functions relate to their bonding functions with respect to indecomposability and connectedness. We show that if such an inverse limit is decomposable then for some n, the Mahavier product of its first n bonding functions is decomposable. It was shown in [3] that if the graphs of bonding functions of a Mahavier-product or inverse limit are pseudoarcs then it is disconnected. We show that a Mahavier-product whose bonding functions have indecomposable graphs can be connected. We also show that the full projection property is not a necessary condition for an indecomposable inverse limit.

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 theorem

Criticality of switching classes of reversible 2-structures labeled by an Abelian group

Let V be a finite vertex set and let (A, +) be a finite abelian group. An A-labeled and reversible 2-structure defined on V is a function g : (V x V) {(v, v) : v is an element of V} -> A such that for distinct u, v is an element of V, g(u, v) = -g (v, u). The set of A-labeled and reversible 2-structures defined on V is denoted by L(V, A). Given g is an element of Y(V, A), a subset X of V is a clan of g if for any x, y is an element of X and v is an element of V\X, g(x, v) = g(y, v). For example, empty set, V and {v} (for v E V) are clans of g, called trivial. An element g of Y(V, A) is primitive if V >= 3 and all the clans of g are trivial. The set of the functions from V to A is denoted by,/(V, A). Given g is an element of L(V, A), with each s is an element of J(V, A) is associated the switch g(s) of g by s defined as follows: given distinct x, y E V, gs (x, y) = s(x) g (x, y) s(y). The switching class of g is {g(s) : s is an element of J(V, A)}. Given a switching class G subset of Y(V, A) and X subset of V, {g((XxX)\,(x,x):x is an element of X}) g is an element of 1 G} is a switching class, denoted by G[X]. Given a switching class G subset of L(V, A), a subset X of V is a clan of G if X is a clan of some g is an element of G. For instance, every X subset of V such that min(X, V X) = 8, there exist u, v is an element of V such that u not equal v and G[V {u, v}] is primitive. Second, we characterize the critical switching classes by using some of the critical digraphs described in [Y. Boudabous and P. Ille, Indecomposability graph and critical vertices of an indecomposable graph, Discrete Math. 309 (2009) 2839-2846].

Decomposing Oriented Graphs into Six Locally Irregular Oriented Graphs

An undirected graph G is locally irregular if every two of its adjacent vertices have distinct degrees. We say that G is decomposable into k locally irregular graphs if there exists a partition $$E_1 \cup E_2 \cup \cdots \cup E_k$$E1źE2źźźEk of the edge set E(G) such that each $$E_i$$Ei induces a locally irregular graph. It was recently conjectured by Baudon et al. that every undirected graph admits a decomposition into at most three locally irregular graphs, except for a well-characterized set of indecomposable graphs. We herein consider an oriented version of this conjecture. Namely, can every oriented graph be decomposed into at most three locally irregular oriented graphs, i.e. whose adjacent vertices have distinct outdegrees? We start by supporting this conjecture by verifying it for several classes of oriented graphs. We then prove a weaker version of this conjecture. Namely, we prove that every oriented graph can be decomposed into at most six locally irregular oriented graphs. We finally prove that even if our conjecture were true, it would remain NP-complete to decide whether an oriented graph is decomposable into at most two locally irregular oriented graphs.

Indecomposability graph and critical vertices of an indecomposable graph

Given a directed graph G = ( V , A ) , the induced subgraph of G by a subset X of V is denoted by G [ X ] . A subset X of V is an interval of G provided that for a , b ∈ X and x ∈ V ∖ X , ( a , x ) ∈ A if and only if ( b , x ) ∈ A , and similarly for ( x , a ) and ( x , b ) . For instance, 0̸ , V and { x } , x ∈ V , are intervals of G , called trivial intervals. A directed graph is indecomposable if all its intervals are trivial, otherwise it is decomposable. Given an indecomposable directed graph G = ( V , A ) , a vertex x of G is critical if G [ V ∖ { x } ] is decomposable. An indecomposable directed graph is critical when all its vertices are critical. With each indecomposable directed graph G = ( V , A ) is associated its indecomposability directed graph Ind ( G ) defined on V by: given x ≠ y ∈ V , ( x , y ) is an arc of Ind ( G ) if G [ V ∖ { x , y } ] is indecomposable. All the results follow from the study of the connected components of the indecomposability directed graph. First, we prove: if G is an indecomposable directed graph, which admits at least two non critical vertices, then there is x ∈ V such that G [ V ∖ { x } ] is indecomposable and non critical. Second, we characterize the indecomposable directed graphs G which have a unique non critical vertex x and such that G [ V ∖ { x } ] is critical. Third, we propose a new approach to characterize the critical directed graphs.

Critically indecomposable graphs

An undirected graph G = ( V , E ) with a specific subset X ⊂ V is called X -critical if G and G ( X ) , induced subgraph on X , are indecomposable but G ( V − { w } ) is decomposable for every w ∈ V − X . This is a generalization of critically indecomposable graphs studied by Schmerl and Trotter [J.H. Schmerl, W.T. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures, Discrete Mathematics 113 (1993) 191–205] and Bonizzoni [P. Bonizzoni, Primitive 2-structures with the ( n − 2 ) -property, Theoretical Computer Science 132 (1994) 151–178], who deal with the case where X is empty. We present several structural results for this class of graphs and show that in every X -critical graph the vertices of V − X can be partitioned into pairs ( a 1 , b 1 ) , ( a 2 , b 2 ) , … , ( a m , b m ) such that G ( V − { a j 1 , b j 1 , … , a j k , b j k } ) is also an X -critical graph for arbitrary set of indices { j 1 , … , j k } . These vertex pairs are called commutative elimination sequence. If G is an arbitrary indecomposable graph with an indecomposable induced subgraph G ( X ) , then the above result establishes the existence of an indecomposability preserving sequence of vertex pairs ( x 1 , y 1 ) , … , ( x t , y t ) such that x i , y i ∈ V − X . As an application of the commutative elimination sequence of an X -critical graph we present algorithms to extend a 3-coloring (similarly, 1-factor) of G ( X ) to entire G .

Indecomposable laplacian integral graphs

A graph that can be constructed from isolated vertices by the operations of union and complement is decomposable. Every decomposable graph is Laplacian integral. i.e., its Laplacian spectrum consists entirely of integers. An indecomposable graph is not decomposable. The main purpose of this note is to demonstrate the existence of infinitely many indecomposable Laplacian integral graphs.

Les graphes 2-reconstructibles indécomposables

A (direct) graph G is 2-reconstructible if any graph H obtained from G by reversing the orientation of some of its directed pairs, chosen arbitrarily, is isomorphic to G. Let G be a 2-reconstructible indecomposable graph, with r directed pairs. It follows from the results of Boussairi and Chaichaa (2003) that G has at least 2 r + 1 vertices. In this Note, we determine, in terms of r, the minimum number of vertices of G. To cite this article: A. Boussaïri, A. Chaïchaâ, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Some Algorithms on Conditionally Critical Indecomposable Graphs

Abstract Let X be a subset of vertices of an undirected graph G = ( V , E ) . G is X-critical if it is indecomposable and its induced subgraph on X vertices is also indecomposable, but every induced subgraphs on V − { w } is decomposable for w ∈ V − X . We present an algorithm to construct an elimination sequence on these graphs preserving critical indecomposability. We also consider two algorithmic problems over X-critical graphs: perfect matching and 3-coloring. It is shown that if a perfect matching for X exists then the same exists for G and it can be computed in O ( n + m log n ) . Also G is shown to be 3-colorable if it is free from fully odd K 4 and G [ X ] is 3-colorable. The time complexity of the 3-coloring algorithm is O ( n ( n + m ) ) .

A Description of Claw-Free Perfect Graphs

It is known that all claw-free perfect graphs can be decomposed via clique-cutsets into two types of indecomposable graphs respectively called elementary and peculiar (1988, V. Chvátal and N. Sbihi, J. Combin. Theory Ser. B 44 , 154–176). We show here that every elementary graph is made up in a well-defined way of a line-graph of bipartite graph and some local augments consisting of complements of bipartite graphs. This yields a complete description of the structure of claw-free Berge graphs and a new proof of their perfectness.

Minimal indecomposable graphs

Let G = ( V, E) be a graph, a subset X of V is an interval of G whenever for a, b ∈ X and x ∈ V − X, ( a, x) ∈ E (resp. ( x, a) ∈ E) if and only if ( b, x) ∈ E (resp. ( x, b) ∈ E). For instance, 0, x, where x ∈ V, and V are intervals of G, called trivial intervals. A graph G is then said to be indecomposable when all of its intervals are trivial. In the opposite case, we will say that G is decomposable. We now introduce the minimal indecomposable graphs in the following way. Given an indecomposable graph G = ( V, E) and vertices x 1, …, x k of G, G is said to be minimal for x 1, …, x k whenever for every proper subset W of V, if x 1, …, x k ∈ W and if | W| ⩾ 3, then the induced subgraph G( W) of G is decomposable. In this paper, we characterize the minimal indecomposable graphs for one or two vertices and we describe in a more precise manner the minimal indecomposable symmetric graphs, posets and tournaments.

Indecomposable graphs

Let G = ( V, E) be a finite directed graph. A subset X of V is an interval of G if for a, b ∈ X and x ∈ V − X, we have ax ∈ E (resp. xa ∈ E) if and only if bx ∈ E (resp. xb ∈ E). So ∅, V and every singleton are intervals of G (called trivial intervals). The graph G is said to be indecomposable if every interval is trivial. In this work, we study the induced subgraphs of an indecomposable graph which are also indecomposable. In particular, we prove: Theorem. Let G = ( V, E) be an indecomposable graph and let X be a subset of V such that | X| ⩾ 3, | V − X| ⩾ 6 and G( X) is indecomposable. Then there is a subset Y of V fulfilling X ⊆ Y, | V − Y| = 2 and G( Y) is indecomposable.

Indecomposable switching graph with the number of verticesN log2 N−9/4N

We propose a new algorithm for constructing switching graphs with many entries and few internal vertices.

Minimal cyclic-4-connected graphs

A theory of cyclic-connectivity is developed, matroid dual to the standard vertex-connectivity. The cyclic- 4 4 -connected graphs minimal under the elementary operations of single-edge deletion or contraction and removal of a trivalent vertex are classified. These turn out to belong to three simple infinite families of indecomposable graphs, or to be decomposable into constituent subgraphs which themselves belong to three simple infinite families. This is modeled after W. T. Tutte’s theorem classifying the minimal 3 3 -connected graphs under single-edge deletion or contraction as forming the single infinite family of "wheels." Such theorems serve two main purposes: (1) illustrating the structure of graphs in the class by isolating a type of extremal graph, and (2) by providing a set-up so that induction on | E ( G ) | |E(G)| can be carried out effectively within the class.

Pentagon-Generated Trivalent Graphs with Girth 5

The terminology of [1] will be assumed in what follows. LetPb(G) stand for the set of pentagons in the graphG.Call a graphpentagongeneratedwhen it is the union of its contained pentagons. LetP5,3be the class of connected trivalent pentagon-generated graphs with girth 5. These graphs form a family including the Petersen graph and the graph of the dodecahedron. They are studied here and completely classified in terms of a decomposition which all but some specifically determined indecomposable graphs admit.Assume henceforth thatH∈P5,3. LetEk(H) be the set of edges in exactlyk∈ 0 pentagons ofH. ClearlyEk(H) = 0 ifk≠ 1, 2, 3, 4 and |E1(H) ∩E(P)r ≦ 2, for allP∈P5(H).P∈P5(H) issingularwhen |E1(H) ∩E(P)r = 2,.