LetA be an n-square non-negative matrix. If A contains no s\times t zero submatrix, where s + t = n, then it is called fully indecomposable. Also, a graph G is said to be fully indecomposable if its adjacency matrix is fully indecomposable. In this paper we provide some necessary and sucient conditions for a graph to be fully indecomposable. Among other results we prove that a regular connected graph is fully indecomposable if and only if it is not bipartite.  

Aaghabali, Mehdi

Akbari, Saieed

Ariannejad, Masoud

Tajfirouz, Zakeieh

English

Contributions to Discrete Mathematics (E-Journal, University of Calgary)

Volume 11, Number 2, Pages 1–8ISSN 1715-0868FULLY INDECOMPOSABLE AND NEARLYDECOMPOSABLE GRAPHSM. AAGHABALI, S. AKBARI, M. ARIANNEJAD, AND Z. TAJFIROUZAbstract. Let A be an n-square non-negative matrix. If A containsno s × t zero submatrix, where s + t = n, then it is called fully inde-composable. Moreover, a graph G is said to be fully indecomposable ifits adjacency matrix is fully indecomposable. In this paper we providesome necessary and sufficient conditions for a graph to be fully indecom-posable. Among other results we prove that a regular connected graphis fully indecomposable if and only if it is not bipartite.1. IntroductionLet G be a graph of order n with the vertex set V (G) = {v1, . . . , vn}.The adjacency matrix of G is an n × n matrix A = [aij ] indexed by thevertex set, where aij = 1, when there is an edge between vi and vj in G andaij = 0, otherwise. By the permanent of any n× n matrix A = [aij ] over acommutative ring we meanper(A) =∑σa1σ(1)a2σ(2) . . . anσ(n),where the summation is over all permutations over {1, . . . , n}. For a graphG, i(G) denotes the number of isolated vertices of G and d(v) denotes thedegree of the vertex v ∈ V (G). For a vertex v we denote by N(v) the set ofall neighbors of vertex v in G; for a subset S ⊆ V (G) we denote by N(S)the set of all neighbors of vertices of S in G. For an n× n square matrixA and integers 1 ≤ i, j ≤ n, A(i|j) denotes the matrix obtained from A byremoving the i-th row and the j-th column. A subgraph H of G is calledspanning subgraph if V (H) = V (G). Furthermore, a spanning subgraph H ofG is called a {1, 2}-factor if every component of H is either a path of order2 or a cycle. We denote by Kn the complete graph of order n that is a graphwith n pairwise adjacent vertices. A transversal in a matrix is a set D of itsentries containing exactly one entry of each row and each column. We sayReceived by the editors November 6, 2014, and in revised form May 28, 2015.2000 Mathematics Subject Classification. 05C50, 05C70, 15A15.Key words and phrases. Permanent, Partly Decomposable, Fully Indecomposable,Factor.The research of the second author was in part supported by a grant from IPM(No.93050214).c©2017 University of Calgary12 M. AAGHABALI, S. AKBARI, M. ARIANNEJAD, AND Z. TAJFIROUZthat D is non-zero if all of its elements are non-zero. It is not hard to checkthat every {1, 2}-factor of a graph is equivalent to a non-zero transversal ofits adjacency matrix.Recall that a matrix A of order n is called partly decomposable if it con-tains an s × t zero submatrix, where s + t = n. In other words, a matrixA is partly decomposable if there exist permutation matrices P and Q suchthatPAQ =[B C0 D],where the zero block is of size s × (n − s). If a matrix contains no s × tzero submatrix, where s + t = n, then it is called fully indecomposable.We say that a graph G is partly decomposable (fully indecomposable) if itsadjacency matrix is partly decomposable (fully indecomposable). Finally,a graph G is said to be nearly decomposable if it is a fully indecomposablegraph but the graph obtained from G by deleting every edge is a partlydecomposable graph. There is research on the fully indecomposable and thenearly decomposable non-negative matrices. Naturally, when we restrict ourattention to some special kind of matrices, we may find further properties. Inthis paper we consider the symmetric (0, 1)-matrices and provide sufficientand necessary conditions for this kind of matrices to be fully indecomposable.As an observation we have the following:Remark 1.1. A graph G is partly decomposable if one of the followingholds:(i) G is not connected;(ii) G is a bipartite graph;(iii) G has a pendant edge.Assume that G is a fully indecomposable graph of order n with the adja-cency matrix A. By the lower bound for the permanent given in [4, p.57],we have per(A) ≥ max{r1, . . . , rn}, where ri =∑nj=1 aij , for i = 1, . . . , n;thus we have per(A) ≥ 2. Therefore every graph with per(A) < 2 is a partlydecomposable graph. Now, let G be a fully indecomposable graph with theadjacency matrix A and per(A) = 2. Then ri = 2 for i = 1, . . . , n, i.e., Gshould be a cycle. It is easy to see that every even cycle is partly decom-posable. Thus every fully indecomposable graph with per(A) = 2 is an oddcycle, which is also a nearly decomposable graph.Assume that G is a graph with the adjacency matrix A. It is well-knownthat per(A) counts the number of {1, 2}-factors of G in which every {1, 2}-factor containing r cycles is counted 2r times. Now, if G is a fully inde-composable graph, then since per(A) ≥ 2, we obtain that G contains a{1, 2}-factor.FULLY INDECOMPOSABLE AND NEARLY DECOMPOSABLE GRAPHS 32. Fully Indecomposable GraphsThe main goal of this section is providing some necessary and sufficientconditions for a graph to be fully indecomposable. We start this section byrecalling a theorem of this nature for a non-negative matrix:Theorem 2.1 ([4, p.38]). Let A be a non-negative n× n matrix with n ≥ 2.Then A is fully indecomposable if and only if per(A(i|j)) > 0 for all i andj.Before stating our first result we need to recall the following useful theo-rem:Theorem 2.2. [4, p.31] Let A be an n× n non-negative matrix. Thenper(A) = 0 if and only if A contains an s × t zero submatrix such thats+ t = n+ 1.Now we are in a position to prove the following theorem:Theorem 2.3. Let G be a connected graph of order n with the adjacencymatrix A. Then for any i, j, with 1 ≤ i, j ≤ n, per(A(i|j)) > 0 if and onlyif per(A(i|i)) > 0.Proof. One direction is clear. For the other let per(A(i|i)) > 0, for i =1, . . . , n, and assume there are two indices i and j such that per(A(i|j)) = 0.By Theorem 2.2, A(i|j) has an s × t zero submatrix C such that s + t =(n − 1) + 1. Assume that the rows and columns of C are indexed by X ={i1, . . . , is} and Y = {j1, . . . , jt}, respectively. If i /∈ Y or j /∈ X, then byTheorem 2.2, per(A(i|i)) = 0 or per(A(j|j)) = 0, a contradiction. So assumethat i ∈ Y and j ∈ X. Now, since G is connected we have X ∩ Y 6= ∅.Therefore there exists an integer 1 ≤ k ≤ n such that k /∈ X ∪ Y . ThusA(k|k) contains C as a zero submatrix which together with Theorem 2.2imply that per(A(k|k)) = 0, a contradiction.This completes the proof of thetheorem. Remark 2.4. The condition of the connectivity in the previous theorem isnot superfluous, as this counterexample will show. Consider the graph Gobtained by the disjoint union of K3 and K4 and let A be the adjacencymatrix of G. It is not hard to verify that A satisfies per(A(i|i)) > 0, fori = 1, . . . , 7. But by the first part of Remark 1.1, since G is not a connectedgraph, A is a partly decomposable matrix.Before stating our next result we need to recall two following theorems.The first one is a celebrated theorem due to Tutte.Theorem 2.5 ([5]). A graph G has a {1, 2}-factor if and only if i(G \S) ≤|S| for every subset S ⊆ V (G).Theorem 2.6 ([3, p.216] and [1, p.253]). Let G be a graph. Then |N(S)| ≥|S| for all independent subset S ⊆ V (G) if and only if G has a {1, 2}-factor.4 M. AAGHABALI, S. AKBARI, M. ARIANNEJAD, AND Z. TAJFIROUZThe following theorem provides some necessary and sufficient conditionsfor a connected graph to be fully indecomposable.Theorem 2.7. Let G be a connected graph. Then the following statementsare equivalent(i) G is a fully indecomposable graph;(ii) i(G \ S) < |S| for all non-empty subset S of vertices;(iii) The graph obtained from G by removing of any vertex contains a{1, 2}-factor;(iv) |N(S)| > |S| for all non-empty independent subset S ⊂ V (G).Proof. (i)⇒ (ii): Assume that G is a fully indecomposable graph. Then Gcontains some {1, 2}-factor, and by Theorem 2.5 we have i(G \ S) ≤ |S| forevery subset S ⊆ V (G). By the contrary suppose that there exists a non-empty subset S ⊆ V (G) such that i(G \S) = |S| and let L denote the set ofall isolated vertices of G\S. Then the submatrix with rows corresponding tothe vertices of L and columns corresponding to V (G)\S is a zero submatrixof size i(G \ S)× (n− |S|), which is a contradiction.(ii) ⇒ (iii): Assume that i(G \ S) < |S| for all non-empty S ⊆ V (G).We prove that for any vertex v ∈ V (G), G′ = G \ {v} contains a {1, 2}-factor. Using Theorem 2.5, assume that S′ ⊆ V (G′) is arbitrary. Then byassumption i(G′ \ S′) = i(G \ (S′ ∪ {v})) is less than |S′| + 1. This yieldsthe result.(iii)⇒ (iv): See Lemma 6.2.1 in [3, p.217].(iv) ⇒ (i): Assume that for every non-empty independent S ⊂ V (G),|N(S)| > |S|. We will show that G is fully indecomposable. By the con-trary assume that G is not fully indecomposable. Then one can find a zerosubmatrix of size s × t, where s + t = n. Thus there exist two subsetsX = {x1, . . . , xs} and Y = {y1, . . . , yt} such that there is no edge betweenX and Y. Since G is connected, S = X∩Y 6= ∅. Clearly S is an independentsubset in which N(S) ⊆ L = V (G) \ (X ∪Y ) and |L| = |S|, a contradiction.This implies that G is fully indecomposable. Note that by Theorem 2.1 one can see that a (0, 1)-matrix A is fully in-decomposable if and only if every 1 in A belongs to a non-zero transversaland every 0 of A belongs to a transversal in which all other elements are1. Therefore adding a new edge to a fully indecomposable graph strictlyincreases the number of {1, 2}-factors of G and removing any edge from Gstrictly decreases the number of {1, 2}-factors. Thus if G is a fully inde-composable graph, then for any edge e of G there exists a {1, 2}-factor of Gcontaining e. The following result was proved by S. Zhou and H. Zhang:Theorem 2.8 ([7]). Let G be a connected non-bipartite graph. Then everyedge of a graph G lies in a {1, 2}-factor if and only if i(G \ S) < |S|, forevery non-empty subset S ⊆ V (G).By Remark 1.1, we know that every fully indecomposable graph is con-nected and non-bipartite. Thus, after combining Theorem 2.7 with theFULLY INDECOMPOSABLE AND NEARLY DECOMPOSABLE GRAPHS 5previous result, we find the following consideration of fully indecomposablegraphs.Theorem 2.9. Let G be a connected graph. Then G is fully indecomposableif and only if G is non-bipartite and every edge of G lies in a {1, 2}-factor.Moreover, we can deduce the following corollary due to Berge [2]:Corollary 2.10. Let G be a connected graph. Then |N(S)| > |S| for allnon-empty S ⊂ V (G) if and only if G is non-bipartite and for each edge eof G, there exists a {1, 2}-factor in G containing e.The previous corollary is a generalization of the following result:Theorem 2.11 ([1, p.20]). Let G be a bipartite graph with bipartition (A,B).If |N(S)| > |S| for all S ⊂ V (G), then for each edge e of G, G has a perfectmatching containing e that saturates A.The following theorem introduces a family of fully indecomposable graphs:Theorem 2.12. Let k be a positive integer. Then a k-regular connectedgraph G is fully indecomposable if and only if it is not bipartite.Proof. One direction is clear, so we provide the other. By Theorem 2.7, Gis fully indecomposable if and only if for any vertex v ∈ V (G), the graphG′ = G\{v} contains a {1, 2}-factor. Note that G′ is a graph of order n−1,in which exactly k vertices have degree k − 1 and n − 1 − k vertices havedegree k. By Theorem 2.5, assume that S ⊆ V (G′) and S contains t verticesof degree k − 1 and denote by L the set of all isolated vertices of G′ \ S.Then |L| = i(G′ \ S). Let L contain r vertices of degree k − 1 in G′. It isclear that 0 ≤ t, r ≤ k. Thus the number of incident edges with L in G′ is(2.1) k|L| − rand the number of incident edges with S in G′ is at most k|S| − t. Thus wefind thatk|L| − r ≤ k|S| − t,or equivalently,|L| ≤ |S|+ r − tk.We claim that r − t < k. If not, then r = k, t = 0, and N(v) ⊆ L. Thuswe have |L| ≤ |S| + 1. Suppose now that |L| = |S| + 1. In this case, byEquation (2.1) the number of incident edges with L in G′ is k|S|. HenceS is an independent set. Now, since G is a connected graph, we concludethat V (G) = S ∪ {v} ∪ L and so G is a bipartite graph with two parts:(S ∪ {v}, L). This is a contradiction, and completes the proof. The following result provides some information about the size of someindependent set of vertices with certain degrees sequence in partly decom-posable graphs:6 M. AAGHABALI, S. AKBARI, M. ARIANNEJAD, AND Z. TAJFIROUZTheorem 2.13. Let G be a connected graph of order n. If G is partlydecomposable, then G contains an independent set S of size k, for someinteger 1 ≤ k ≤ n/2 satisfying k ≥ maxv∈S d(v).Proof. Let A be the adjacency matrix of G. Then A contains an s× t zerosubmatrix with s + t = n. Thus there are two subsets X = {x1, . . . , xs}and Y = {y1, . . . , yt}, such that there is no edge between X and Y. SinceG is connected, S = X ∩ Y 6= ∅. Now suppose that S = {z1, . . . , zk}.Clearly 1 ≤ k ≤ n/2 and S is an independent set of G. Furthermore,N(S) ⊆ L = V (G) \ (X ∪ Y ) and |L| = k. This completes the proof. Let G be a graph with a fully indecomposable spanning subgraph H. ThenG should be a fully indecomposable graph , which gives us the followingremark.Remark 2.14. If G is a graph of order 2n+ 1 with minimum degree δ suchthat δ ≥ n + 1, then G is a fully indecomposable graph. To see this, notethat using Dirac’s Theorem [6, p.288] with δ ≥ n + 1 implies that G hasa Hamiltonian cycle. However, every odd cycle is a fully indecomposablegraph.This fact does not hold for the graphs of order 2n and δ ≥ n. To showthis we introduce a graph of order 2n with minimum degree n that is partlydecomposable. Let {v1, . . . , vn} be the vertex set of Kn. Denote by G =Kn ∨Kn the graph obtained from Kn by adding new vertices {w1, . . . , wn}and edges {wi, vj} for i, j = 1, . . . , n. Consider the submatrix B of theadjacency matrix of G whose rows and columns are indexed by the vertices{w1, . . . , wn}. Since {w1, . . . , wn} is an independent set of vertices in G, Bis a zero submatrix of size n × n. Thus G is a partly decomposable graphwith minimum degree n.3. Nearly Decomposable GraphsWe now study some properties of the nearly decomposable graphs. Zhouand Zhang in [7] studied the graphs G whose every edge lies in a {1, 2}-factor, with the further condition that for all edges e of G, the graph G− eobtained from deleting the edge e of G does not have that every edge lies in a{1, 2}-factor. The graphs with this property are called minimal 2-matching-covered graphs. The following lemma describes the nearly decomposablegraphs in view of minimal 2-matching-covered graphs.Lemma 3.1. Let G be a connected graph. Then G is nearly decomposableif and only if G is a non-bipartite minimal 2-matching-covered graph.Proof. Let G be a nearly decomposable graph. Hence by definition, G isa fully indecomposable graph in which the graph obtained by removing ofevery edge is partly decomposable. Combining Remark 1.1 and Theorem 2.9,we find that G is a non-bipartite graph whose every edge lies in a {1, 2}-factor. However, for all edges e of G, the graph G − e obtained from G byFULLY INDECOMPOSABLE AND NEARLY DECOMPOSABLE GRAPHS 7deleting the edge e does not satisfy this property. Thus G is a non-bipartiteminimal 2-matching-covered graph.The converse is clear. This completes the proof. Hence, using the properties of minimal 2-matching-covered graphs andthe previous equivalence, we can obtain the following result:Theorem 3.2. Let G be a nearly decomposable graph. Then:(i) δ(G) = 2, except for G = K4;(ii) Removing every perfect matching of an induced 4-cycle of G yieldsan elementary bipartite graph.Let F(n) denote the set of all fully indecomposable graphs of order n.Clearly, every graph G ∈ F(n) with the minimum number of edges is anearly decomposable graph. Now considerµ := minG∈F(n)per(AG) = per(AG∗),where AG is the adjacency matrix of G.The following theorem states that G∗ is a nearly decomposable graph:Theorem 3.3. Let G∗ be a fully indecomposable graph with the adjacencymatrix AG∗. Assume that µ = per(AG∗), with µ given as above. Then G∗ isa nearly decomposable graph.Proof. First we show that every fully indecomposable graph contains a span-ning nearly decomposable subgraph. Assume that G is a fully indecompos-able graph of order n and denote by F(G) the set of all fully indecomposablespanning subgraphs of G. Let G′ ∈ F(G) be a graph with the minimumnumber of edges. Then G′ is a nearly decomposable spanning subgraph ofG.Now assume that G is a fully indecomposable graph which is not nearlydecomposable. By the above argument we may find a nearly decomposablespanning subgraph of G, say G′, whose number of edges is less than thenumber of edges of G. But by Theorem 2.1, removing any edge of a fullyindecomposable graph strictly decreases the number of {1, 2}-factors. Thuswe find per(AG′) < per(AG). This completes the proof. 4. AcknowledgementsThe authors would like to express their deep gratitude to the referee forcareful reading of the paper and her/his valuable comments. The secondauthor is indebted to the Research Council of Sharif University of Technol-ogy and the School of Mathematics, Institute for Research in FundamentalSciences (IPM) for support.8 M. AAGHABALI, S. AKBARI, M. ARIANNEJAD, AND Z. TAJFIROUZReferences[1] J. Akiyama and M. Kano, Factors and factorizations of graphs, Lecture Notes inMathematics, vol. 2031, Springer, Heidelberg, 2011, Proof techniques in factor theory.[2] C. Berge, Regularisable graphs I, Disc. Math. 23 (1978), 85–89.[3] L. Lova´sz and M. Plummer, Matching theory, North-Holland Mathematics Studies,vol. 121, North-Holland Publishing Co., Amsterdam; Akade´miai Kiado´ (PublishingHouse of the Hungarian Academy of Sciences), Budapest, 1986, Annals of DiscreteMathematics, 29.[4] H. Minc, Permanents, Encyclopedia of Mathematics and its Applications, vol. 6,Addison-Wesley Publishing Co., Reading, Mass., 1978, With a foreword by MarvinMarcus.[5] W. T. Tutte, The 1-factors of oriented graphs, Proc. Amer. Math. Soc. 4 (1953),922–931.[6] D. B. West, Introduction to graph theory, 2 ed., Prentice Hall, Inc., Upper SaddleRiver, NJ, 2001.[7] S. Zhou and H. Zhang, Minimal 2-matching-covered graphs, Discrete Math. 309 (2009),no. 13, 4270–4279.Department of Mathematics, Mobarakeh Branch, Islamic Azad University,Mobarakeh, Isfahan, Iran.E-mail address: m2aghabali@yahoo.comDepartment of Mathematical Sciences, Sharif University of Technology,P.O. Box 11155-9415, Tehran, Iran.School of Mathematics, Institute for Research in Fundamental Sciences(IPM), P.O. Box 19395-5746, Tehran, Iran.E-mail address: s akbari@sharif.eduDepartment of Mathematics, University of Zanjan, P.O. Box 45371-38791,Zanjan, Iran.E-mail address: arian@znu.ac.irDepartment of Mathematics, University of Zanjan, P.O. Box 45371-38791,Zanjan, Iran.E-mail address: z tajfirouz@yahoo.com

Fully Indecomposable and Nearly Decomposable Graphs

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Fully Indecomposable and Nearly Decomposable Graphs

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Contributions to Discrete Mathematics (E-Journal, University of Calgary)