Proceedings of Graph Theory@Georgia Tech, a conference honoring the 50th Birthday of Robin Thomas, May 7-11, 2012 in the Clough Undergraduate Learning Commons.We introduced a new Colin de Verdiere-type invariant \nu(G,\Sigma) for signed graphs. This invariant is closed under taking minors, and characterizes bipartite signed graphs as those signed graphs (G,\Sigma) with \nu(G,\Sigma)\leq 1, and signed graphs with no odd-K_4- and no odd-K^2_3-minor as those signed graphs (G,\Sigma) with \nu(G,\Sigma)\leq 2. In this talk we will discuss this invariant and these results.  Joint work with Marina Arav, Frank Hall, and Zhongshan Li.NSF, NSA, ONR, IMA, Colleges of Sciences, Computing and Engineerin

Van der Holst, Hein

Scholarly Materials And Research @ Georgia Tech

A Colin de Verdiere-Type Invariant and Odd-K4- andOdd-K 23 -Free Signed GraphsHein van der HolstGeorgia State UniversityConference honouring Robin’s 50th birthdayJoint work with Marina Arav, Frank Hall, and Jason LiHein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 1 / 31The parameter µ.DefinitionFor a graph G = (V ,E ) with n vertices, O(G ) is the set of all symmetricn × n matrices A = [ai ,j ] with1 ai ,j < 0 if i and j are adjacent,2 ai ,j = 0 if i 6= j and i and j are non-adjacent3 ai ,i ∈ R for i ∈ V .Definition (Y. Colin de Verdière)µ(G ) is the largest possible nullity (corank) of any A ∈ O(G ) that hasexactly one negative eigenvalue and that satisfies the Strong ArnoldHypothesis (SAH).Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 2 / 31Theorem1 µ(G ) ≤ 3 if and only if G is planar. (Colin de Verdière)2 µ(G ) ≤ 4 if and only if G has a flat embedding. (Conjectured byRobertson, Seymour, Thomas. Proved by Lovász and Schrijver.)No characterization known for graphs G with µ(G ) ≤ 5.Conjecture (Luke Postle)For all graphs H, there exists a cH such that all graphs G with no H-minorhave µ(G ) ≤ cH .Support for this conjecture:1 µ behaves well under clique sums,2 behaves well under adding vertices, and3 is bounded for graphs embedded on a fixed surface. But the problemis how to deal with vortices.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 3 / 31Theorem1 µ(G ) ≤ 3 if and only if G is planar. (Colin de Verdière)2 µ(G ) ≤ 4 if and only if G has a flat embedding. (Conjectured byRobertson, Seymour, Thomas. Proved by Lovász and Schrijver.)No characterization known for graphs G with µ(G ) ≤ 5.Conjectureµ is bounded on an ideal I if and only if I is a proper ideal.Support for this conjecture:1 µ behaves well under clique sums,2 behaves well under adding vertices, and3 is bounded for graphs embedded on a fixed surface. But the problemis how to deal with vortices.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 3 / 31The parameter νDefinitionFor a graph G = (V ,E ) with n vertices, S(G ) is the set of all symmetricn × n matrices A = [ai ,j ] with1 ai ,j 6= 0 if i and j are adjacent,2 ai ,j = 0 if i 6= j and i and j are non-adjacent,3 ai ,i ∈ R for all i ∈ V .Definition (Colin de Verdière)ν(G ) is the largest possible nullity of any positive semidefinite matrixA = [ai ,j ] ∈ S(G ) that satisfies the SAH.Theoremν(G ) is large if and only if tree-width G is large.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 4 / 31Can we generalize these to one graph parameter?Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 5 / 31Signed graphsDefinitionA signed graph is a pair (G ,Σ) with Σ ⊆ E (G ). (We allow parallel edges,but no loops.) Edges in Σ are called odd edges and the other edges arecalled even edges.DefinitionA cycle C in (G ,Σ) is called odd if C has an odd number of odd edges.ExampleA signed graph (G ,Σ) is called bipartite if it has no odd cycles.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 6 / 31Theorem (Zaslavsky)(G ,Σ) and (G ,Ω) have the same set of odd cycle if and only if there is anS ⊆ V (G ) such that Ω = Σ∆δ(S).DefinitionThe operation Σ→ Σ∆δ(S) is called a signature-exchange.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 7 / 31DefinitionFor a signed graph (G ,Σ) with n vertices, S(G ,Σ) is the set of allsymmetric n × n matrices A = [ai ,j ] with1 ai ,j < 0 if i and j are connected by only even edges,2 ai ,j > 0 if i and j are connected by only odd edges,3 ai ,j ∈ R if i and j are connected by even and odd edges,4 ai ,j = 0 if i and j are not adjacent,5 ai ,i ∈ R for all i ∈ V (G ).Example−1 −1 −1−1 −1 −1−1 −1 −1 ∈ S(K3, ∅).Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 8 / 31The inertia setDefinitionThe inertia set I (G ,Σ) of (G ,Σ) is the set of all pairs (p, q) for whichthere exists an A ∈ S(G ,Σ) with exactly p positive and q negativeeigenvalues.Example(0, 1) ∈ I (K3, ∅).TheoremA signature-exchange on (G ,Σ) does not change the inertia set.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 9 / 31Northeast LemmaLemmaIf (G ,Σ) has n vertices, (p, q) ∈ I (G ,Σ), and p + q < n, then(p + 1, q), (p, q + 1) ∈ I (G ,Σ).Shown for graphs by Barrett, Hall, and Loewy.For I (K3, ∅),p1 2 3123qHein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 10 / 31Stable inertia setThe inertia set of a signed graph is difficult to determine. Can we useminors?DefinitionThe stable inertia set I s(G ,Σ) of (G ,Σ) is the set of all pairs (p, q) forwhich there exists an A ∈ S(G ,Σ) that has exactly p positive eigenvaluesand q negative eigenvalues and that have the Strong Arnold Hypothesis.DefinitionT (G ) denotes the set of all symmetric n × n matrices A = [ai ,j ] withai ,j = 0 if i 6= j and i and j are not adjacent.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 11 / 31Stable inertia setThe inertia set of a signed graph is difficult to determine. Can we useminors?DefinitionThe stable inertia set I s(G ,Σ) of (G ,Σ) is the set of all pairs (p, q) forwhich there exists an A ∈ S(G ,Σ) that has exactly p positive eigenvaluesand q negative eigenvalues and that have the Strong Arnold Hypothesis.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 11 / 31Stable inertia setThe inertia set of a signed graph is difficult to determine. Can we useminors?DefinitionThe stable inertia set I s(G ,Σ) of (G ,Σ) is the set of all pairs (p, q) forwhich there exists an A ∈ S(G ,Σ) that has exactly p positive eigenvaluesand q negative eigenvalues and that have the Strong Arnold Hypothesis.DefinitionT (G ) denotes the set of all symmetric n × n matrices A = [ai ,j ] withai ,j = 0 if i 6= j and i and j are not adjacent.DefinitionA matrix A ∈ S(G ,Σ) satisfies the Strong Arnold Hypothesis (SAH) if forall symmetric matrices K , there exists a matrix M ∈ T (G ) such thatxTKx = xTMx for all x ∈ ker(A).Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 11 / 31A =1 −1 −1 −1 −1−1 0 0 0 0−1 0 0 0 0−1 0 0 0 0−1 0 0 0 0 ∈ S(K1,4, ∅)Nullity A is 3, but A does not have SAH.(1, 1) ∈ I (K1,4, ∅)But(1, 1) 6∈ I s(K1,4, ∅).Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 12 / 31TheoremIf (G ,Σ) has n vertices, (p, q) ∈ I s(G ,Σ), and p + q < n, then(p + 1, q), (p, q + 1) ∈ I s(G ,Σ).Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 13 / 31Stable inertia set and subgraphsN includes 0.N2k = {(p, q) ∈ N2 | p + q = k}.N2[k,n] = {(p, q) ∈ N2 | k ≤ p + q ≤ n}.If A,B ⊆ N2, thenA + B = {(p1 + p2, q1 + q2) | (p1, q1) ∈ A, (p2, q2) ∈ B}.TheoremLet (G ,Σ) be a signed graph with n vertices and (H,Σ ∩ E (H)) a signedsubgraph with m vertices. Then I s(H,Σ ∩ E (H)) + N2n−m ⊆ I s(G ,Σ).CorollaryIf (G ,Σ) has n > 0 vertices, then N2[n−1,n] ⊆ Is(G ,Σ) ⊆ I (G ,Σ).Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 14 / 31For (K3, ∅),31 2 312Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 15 / 31Stable inertia set and contracting edgesTheoremLet (G ,Σ) be a signed graph and e ∈ E (G ).1 If e is odd, then I s(G/e,Σ \ {e}) + {(0, 1)} ⊆ I s(G ,Σ).2 If e is even, then I s(G/e,Σ) + {(1, 0)} ⊆ I s(G ,Σ).Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 16 / 31Our friend reappears.What happens if we consider signed graph of the form (G , ∅)?TheoremIf G has n vertices, µ(G ) = max{k ≥ 0 | (n − k − 1, 1) ∈ I s(G , ∅)}.Remember:Theorem1 µ(G ) ≤ 3 if and only if G is planar.2 µ(G ) ≤ 4 if and only if G has a flat embedding.No characterization known for graphs G with µ(G ) ≤ 5.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 17 / 31ν(G ,Σ)DefinitionFor a signed graph (G ,Σ), ν(G ,Σ) is the largest nullity (corank) of anypositive semidefinite matrix A ∈ S(G ,Σ) satisfying the Strong ArnoldHypothesis.Observationν(G ) = max{ν(G ,Σ) | Σ ⊆ E (G )}.ObservationIf G has n vertices, ν(G ,Σ) = max{k ≥ 0 | (n − k , 0) ∈ I s(G ,Σ)}.ProblemCharacterize signed graph (G ,Σ) with ν(G ,Σ) ≤ 1 and those withν(G ,Σ) ≤ 2.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 18 / 31Minor-monotonicity of ν(G ,Σ)DefinitionA signed graph (H,Ω) is a minor of (G ,Σ) if (H,Ω) arises from (G ,Σ) bya series of deletions of vertices and edges, contractions of even edges, andsignature-exchanges.No contraction of odd edges allowed. First signature-exchange to makethe edge even, then contraction.TheoremIf (H,Ω) is a minor of (G ,Σ), then ν(H,Ω) ≤ ν(G ,Σ).Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 19 / 31Bipartite signed graphsTheoremν(G ,Σ) ≤ 1 if and only if (G ,Σ) is bipartite (i.e. has no odd cycles).Proof.ν(C2, {e}) = 2, so (G ,Σ) has no odd cycles.For the converse, suppose (G ,Σ) has no odd cycle.May assume (G ,Σ) is connected.Apply signature-exchange to make all edges even.Perron-Frobenius says that smallest eigenvalue has multiplicity one.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 20 / 31ProblemWhich signed graphs (G ,Σ) have ν(G ,Σ) ≤ 2?Lemmaν(odd-K4) = 3 and ν(odd-C23 ) = 3.LemmaIf ν(G ,Σ) ≤ 2, then (G ,Σ) has no odd-K4 and no odd-C 23 .Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 21 / 31No odd-K4 and no odd-C23 .Example (of no odd-K4 and no odd-C23 .)Signed graphs (G ,Σ) for which there is a vertex v such that (G ,Σ) \ v isbipartite.DefinitionA signed graph (G ,Σ) is almost bipartite if there is a vertex v such that(G ,Σ) \ v is bipartite.ExampleSigned graphs (G ,Σ) such that G can be embedded in the plane with atmost two odd faces.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 22 / 31Figure: PrismHein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 23 / 31Theorem (Gerards)Let (G ,Σ) be a signed graph with no odd-K4- and no odd-C23 -minor.Then at least one of the following holds:1 (G ,Σ) has a 0-, 1-, 2-, or 3-split;2 (G ,Σ) is almost bipartite;3 (G ,Σ) can be embedded in the plane with at most two odd faces;4 (G ,Σ) is equivalent to the prism.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 24 / 31TheoremIf (G1,Σ1) and (G2,Σ2) form a 0-, 1-, or a 2-split of (G ,Σ), thenν(G ,Σ) = max{ν(G1,Σ2), ν(G2,Σ2)}.TheoremIf (G1,Σ1) is a 3-split of (G ,Σ), then ν(G ,Σ) = ν(G1,Σ1).Hence we need to prove that almost bipartite signed graphs, planar signedgraph with two odd faces, and the prism have ν(G ,Σ) ≤ 2.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 25 / 31Almost bipartiteTheoremIf (G ,Σ) is almost bipartite, then ν(G ,Σ) ≤ 2.Proof.May assume (G ,Σ) is 2-connected.Suppose ν(G ,Σ) ≥ 3. Let A ∈ S(G ,Σ) be positive semidefinite withnullity ≥ 3.Can delete a vertex v such that (G ,Σ) \ {v} is connected andbipartite.Let w be any other vertex.Let x ∈ ker(A) be nonzero and with xv = xw = 0. Perron-Frobeniussays restriction of x to (G ,Σ) \ {v} must be either all positive or allnegative.Contradiction.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 26 / 31Planar with two odd facesTheoremIf (G ,Σ) is planar with two odd faces, then ν(G ,Σ) ≤ 2.Proof Idea.Theorem (Epifanov)Any planar graph with two terminal can be reduced to a single edge,where in each reduction none of the terminal is involved.Reductions are: ∆Y -exchanges, suppressing degree two vertices, deletingdegree one and zero vertices, and removal of all but one edge from eachparallel class. Terminals cannot be removed.Dualize this theorem. Odd faces ↔ terminals.ν(G ,Σ) does not decrease under reductions.End up with signed graph (C2, {e}). Has ν(C2, {e}) = 2.Hence ν(G ,Σ) ≤ 2.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 27 / 31PrismTheoremν(prism) = 2.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 28 / 31Theoremν(G ,Σ) ≤ 2 if and only if (G ,Σ) has no odd-K4 and no odd-C 23 .Problem (Open)What about ν(G ,Σ) ≤ 3?Questionν(G ,Σ) is large if and only if (G ,Σ) contains a large super-triangle as aminor.evenodd odd odd oddoddoddoddodd oddoddeven even evenevenevenFigure: SupertriangleHein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 29 / 31Other resultsDefinitionξ(G ,Σ) is the largest nullity of any A ∈ S(G ,Σ) satisfying the SAH.(Without the restriction of positive semidefinite.)We have combinatorial characterization of signed graphs (G ,Σ) withξ(G ,Σ) ≤ 1 and of signed graphs (G ,Σ) with ξ(G ,Σ) ≤ 2.Hein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 30 / 31Happy BirthdayRobinHein van der Holst (Georgia State) Odd-K4- and Odd-K23 -Free Signed Graphs May 11, 2012 31 / 31

A Colin de Verdiere-Type Invariant and Odd-K_4- and Odd-K^2_3-Free Signed Graphs

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A Colin de Verdiere-Type Invariant and Odd-K_4- and Odd-K^2_3-Free Signed Graphs

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