Extensional Acyclic Orientations and Set Graphs

Transcription

1 Extensional Acyclic Orientations and Set Graphs Martin Milanič 1, Romeo Rizzi 2, Alexandru I. Tomescu 2,3 1 University of Primorska 2 Università di Udine 3 University of Bucharest 7 th Slovenian International Conference on Graph Theory Bled 11 June 24, / 21

2 OUTLINE SETS, DIGRAPHS, SET GRAPHS RESULTS ON SET GRAPHS APPLICATIONS 2 / 21

3 SETS, DIGRAPHS, SET GRAPHS RESULTS ON SET GRAPHS APPLICATIONS 3 / 21

4 HEREDITARILY FINITE SETS We assume the axioms of Zermelo-Fraenkel, including Axiom of Extensionality: two sets are equal iff they have the same elements Axiom of Foundation: there are no membership cycles or infinite descending membership chains 4 / 21

5 HEREDITARILY FINITE SETS We assume the axioms of Zermelo-Fraenkel, including Axiom of Extensionality: two sets are equal iff they have the same elements Axiom of Foundation: there are no membership cycles or infinite descending membership chains The standard model: von Neumann s cumulative hierarchy of sets. Its subclass of hereditarily finite sets, HF: HF = i V i, over all i N, V0 = ø, Vi+1 = P(V i ). 4 / 21

6 HEREDITARILY FINITE SETS We assume the axioms of Zermelo-Fraenkel, including Axiom of Extensionality: two sets are equal iff they have the same elements Axiom of Foundation: there are no membership cycles or infinite descending membership chains The standard model: von Neumann s cumulative hierarchy of sets. Its subclass of hereditarily finite sets, HF: HF = i V i, over all i N, V0 = ø, Vi+1 = P(V i ). For example, V1 = {ø} V2 = { ø, {ø} } { } V 3 = ø, {ø}, {{ø}}, {ø, {ø}},... 4 / 21

7 REPRESENTING SETS BY DIGRAPHS { ø, {{ø}}, {ø, {ø}} }

8 REPRESENTING SETS BY DIGRAPHS { ø, {{ø}}, {ø, {ø}} } {{ø}} {ø, {ø}} ø

9 REPRESENTING SETS BY DIGRAPHS { ø, {{ø}}, {ø, {ø}} } {{ø}} {ø, {ø}} {ø} ø

10 REPRESENTING SETS BY DIGRAPHS { ø, {{ø}}, {ø, {ø}} } {{ø}} {ø, {ø}} {ø} ø

11 REPRESENTING SETS BY DIGRAPHS { ø, {{ø}}, {ø, {ø}} } {{ø}} {ø, {ø}} {ø} ø 5 / 21

12 REPRESENTING SETS BY DIGRAPHS { ø, {{ø}}, {ø, {ø}} } {{ø}} {ø, {ø}} {ø} The transitive closure of a set x TrCl(x) := x y x TrCl(y). ø 5 / 21

13 REPRESENTING SETS BY DIGRAPHS { ø, {{ø}}, {ø, {ø}} } {{ø}} {ø, {ø}} {ø} The transitive closure of a set x TrCl(x) := x y x TrCl(y). ø DEFINITION The membership digraph of x is ( ) TrCl(x), {u v u, v TrCl(x) u v}. 5 / 21

14 SET GRAPHS DEFINITION A digraph D is extensional if u v V(G), N + (u) N + (v). 6 / 21

15 SET GRAPHS DEFINITION A digraph D is extensional if u v V(G), N + (u) N + (v). There is a 1-1 correspondence between membership digraphs and extensional acyclic (e.a.) digraphs. 6 / 21

16 SET GRAPHS DEFINITION A digraph D is extensional if u v V(G), N + (u) N + (v). There is a 1-1 correspondence between membership digraphs and extensional acyclic (e.a.) digraphs. MAIN QUESTION Characterize the underlying (undirected) graphs of e.a. digraphs. we consider finite simple (di)graphs; if a graph admits an e.a. orientation, we call it a set graph. 6 / 21

17 SET GRAPHS DEFINITION A digraph D is extensional if u v V(G), N + (u) N + (v). There is a 1-1 correspondence between membership digraphs and extensional acyclic (e.a.) digraphs. MAIN QUESTION Characterize the underlying (undirected) graphs of e.a. digraphs. we consider finite simple (di)graphs; if a graph admits an e.a. orientation, we call it a set graph. We study the implicit graph-theoretic expressive power of HF: a set graph can be represented by the transitive closure from which it originates (e.g. in {log}, CLP(SET ), Referee) 6 / 21

18 SETS, DIGRAPHS, SET GRAPHS RESULTS ON SET GRAPHS APPLICATIONS 7 / 21

19 BASIC PROPERTIES Every set graph is connected (since every e.a. orientation has a unique sink) There are connected graphs which are not set graphs: e.g., the claw, K 1,3????? 8 / 21

20 BASIC PROPERTIES Every set graph is connected (since every e.a. orientation has a unique sink) There are connected graphs which are not set graphs: e.g., the claw, K 1,3???? PROPOSITION If G is a set graph, then for every X V(G), G X has at most 2 X connected components.? The only trees which are set graphs are paths. 8 / 21

21 9 / 21

22 BASIC PROPERTIES THEOREM If G has a Hamiltonian path, then G is a set graph. 10 / 21

23 BASIC PROPERTIES THEOREM If G has a Hamiltonian path, then G is a set graph. 10 / 21

24 BASIC PROPERTIES THEOREM If G has a Hamiltonian path, then G is a set graph. x z y 10 / 21

25 BASIC PROPERTIES THEOREM If G has a Hamiltonian path, then G is a set graph. x z y Not all set graphs have a Hamiltonian path: 10 / 21

26 BASIC PROPERTIES THEOREM If G has a Hamiltonian path, then G is a set graph. x z y COROLLARY Given G a complete multipartite graph, then G is a set graph if and only if G has a Hamiltonian path. COROLLARY The tree-width of set graphs is not bounded by a constant. 10 / 21

27 RECOGNIZING SET GRAPHS IS NP-COMPLETE Let S(G) be obtained from G by subdividing each edge once. THEOREM G has a Hamiltonian path if and only if S(G) is a set graph. 11 / 21

28 RECOGNIZING SET GRAPHS IS NP-COMPLETE Let S(G) be obtained from G by subdividing each edge once. THEOREM G has a Hamiltonian path if and only if S(G) is a set graph. 11 / 21

29 RECOGNIZING SET GRAPHS IS NP-COMPLETE Let S(G) be obtained from G by subdividing each edge once. THEOREM G has a Hamiltonian path if and only if S(G) is a set graph. 11 / 21

30 RECOGNIZING SET GRAPHS IS NP-COMPLETE Let S(G) be obtained from G by subdividing each edge once. THEOREM G has a Hamiltonian path if and only if S(G) is a set graph. 11 / 21

31 RECOGNIZING SET GRAPHS IS NP-COMPLETE Let S(G) be obtained from G by subdividing each edge once. THEOREM G has a Hamiltonian path if and only if S(G) is a set graph. 11 / 21

32 RECOGNIZING SET GRAPHS IS NP-COMPLETE Let S(G) be obtained from G by subdividing each edge once. THEOREM G has a Hamiltonian path if and only if S(G) is a set graph. For the reverse implication, prove that any e.a. orientation of S(G) has a directed path passing through all vertices of G. It remains NP-complete for bipartite graphs, of maximum degree / 21

33 MORE ON COMPLEXITY By the previous reduction, this is NP-complete. IN: A graph G and S V(G). OUT: Does G admit an e.a.o. whose set of sources includes S? 12 / 21

34 MORE ON COMPLEXITY By the previous reduction, this is NP-complete. IN: A graph G and S V(G). OUT: Does G admit an e.a.o. whose set of sources includes S? By a more involved reduction, also this is NP-complete. IN: A graph G and S V(G). OUT: Does G admit an e.a.o. whose set of sources is precisely S? 12 / 21

35 MORE ON COMPLEXITY By the previous reduction, this is NP-complete. IN: A graph G and S V(G). OUT: Does G admit an e.a.o. whose set of sources includes S? By a more involved reduction, also this is NP-complete. IN: A graph G and S V(G). OUT: Does G admit an e.a.o. whose set of sources is precisely S? However, these problems can be stated in MSOL. By Courcelle s theorem, we get: THEOREM Recognizing set graphs and the above problem can be solved in linear time on graphs of bounded tree-width. 12 / 21

36 CONNECTED CLAW-FREE GRAPHS ARE SET GRAPHS THEOREM If G is a connected claw-free graph, then G is a set graph; an e.a. orientation can be found in polynomial time. 13 / 21

37 CONNECTED CLAW-FREE GRAPHS ARE SET GRAPHS THEOREM If G is a connected claw-free graph, then G is a set graph; an e.a. orientation can be found in polynomial time. Proof (algorithm) idea: in any acyclic orientation with a unique sink of G, there can be no collision among three vertices x y z start with a good acyclic orientation with a unique sink use claw-freeness to resolve each collision, by reversing at most two arcs w 13 / 21

38 MORE ON CLAW-FREENESS Claw-free graphs = the largest hereditary class of graphs every connected member of which is a set graph. 14 / 21

39 MORE ON CLAW-FREENESS Claw-free graphs = the largest hereditary class of graphs every connected member of which is a set graph. apple K 2,3 dart co-(k 3 + 2K 1 ) THEOREM If G is (apple, K 2,3, dart, co-(k 3 + 2K 1 ))-free and connected, then G is a set graph if and only if G is claw-free. this is the largest hereditary class with this property 14 / 21

40 CLOSURE PROPERTIES Set graphs are not closed under taking induced subgraphs. Set graphs are closed under: substitution (hence under adding a dominant vertex, or adding a true twin) x H suppressing a cut vertex G G x G G 15 / 21

41 UNICYCLIC SET GRAPHS Recognizing set graphs is easy the extremal values of the cyclomatic number (i.e., trees, complete graphs). THEOREM A unicyclic graph is a set graph if and only if it is a jellyfish graph. Moreover, an e.a. orientation of such a graph can be found in linear time. P C P A jellyfish graph; any of the paths P, P, P can have length 0 P 16 / 21

42 SETS, DIGRAPHS, SET GRAPHS RESULTS ON SET GRAPHS APPLICATIONS 17 / 21

43 SIMPLER PROOFS FOR CLAW-FREE GRAPHS / AUTOMATIC PROOF VERIFICATION By working with an acyclic orientation with a unique sink, (e.g., extensional) The square of any connected claw-free graph is vertex-pancyclic [Matthews, Sumner, 1984] here we avoid resorting to a general result that in G 2, vertex-pancyclicity Hamiltonicity [Fleischner, 1976] 18 / 21

44 SIMPLER PROOFS FOR CLAW-FREE GRAPHS / AUTOMATIC PROOF VERIFICATION By working with an acyclic orientation with a unique sink, (e.g., extensional) The square of any connected claw-free graph is vertex-pancyclic [Matthews, Sumner, 1984] here we avoid resorting to a general result that in G 2, vertex-pancyclicity Hamiltonicity [Fleischner, 1976] Any connected claw-free graph of even order has a perfect matching [Sumner, 1974] 18 / 21

45 SIMPLER PROOFS FOR CLAW-FREE GRAPHS / AUTOMATIC PROOF VERIFICATION By working with an acyclic orientation with a unique sink, (e.g., extensional) The square of any connected claw-free graph is vertex-pancyclic [Matthews, Sumner, 1984] here we avoid resorting to a general result that in G 2, vertex-pancyclicity Hamiltonicity [Fleischner, 1976] Any connected claw-free graph of even order has a perfect matching [Sumner, 1974] These proofs are currently being formalized in the automatic proof-checker Referee. representing a connected claw-free graph by a transitive set required the minimal formalism. 18 / 21

46 VERTEX IDENTIFICATION Practical application: discriminating codes for graphs [Karpovsky, Chakrabarty, Levitin, 1998], [Charon, Cohen, Hudry, Lobstein, 2008] 19 / 21

47 VERTEX IDENTIFICATION Practical application: discriminating codes for graphs [Karpovsky, Chakrabarty, Levitin, 1998], [Charon, Cohen, Hudry, Lobstein, 2008] Given a digraph D, we say that a subset C V(D) is a discriminating code if u v V(D), N + (u) C N + (v) C. 19 / 21

48 VERTEX IDENTIFICATION Practical application: discriminating codes for graphs [Karpovsky, Chakrabarty, Levitin, 1998], [Charon, Cohen, Hudry, Lobstein, 2008] Given a digraph D, we say that a subset C V(D) is a discriminating code if u v V(D), N + (u) C N + (v) C. A digraph D admits a discriminating code if and only if D is extensional. 19 / 21

49 VERTEX IDENTIFICATION Practical application: discriminating codes for graphs [Karpovsky, Chakrabarty, Levitin, 1998], [Charon, Cohen, Hudry, Lobstein, 2008] Given a digraph D, we say that a subset C V(D) is a discriminating code if u v V(D), N + (u) C N + (v) C. A digraph D admits a discriminating code if and only if D is extensional. Useful, e.g., in fault detection in multiprocessor systems. 19 / 21

50 CONCLUSIONS Set graphs arise from set theory, but have many (practical) applications Recognizing set graphs is NP-complete but solvable in linear time on graphs of bounded tree-width Set graphs generalize: graphs with a Hamiltonian path connected claw-free graphs (an e.a. orientation can be found in poly-time) 20 / 21

51 CONCLUSIONS Set graphs arise from set theory, but have many (practical) applications Recognizing set graphs is NP-complete but solvable in linear time on graphs of bounded tree-width Set graphs generalize: graphs with a Hamiltonian path connected claw-free graphs (an e.a. orientation can be found in poly-time) Worth studying: allowing the orientation to have cycles, and using an extensional-like criterion for irredundacy; e.g., bisimilarity and hypersets: recognizing graphs which admit an orientation without distinct bisimilar vertices is also NP-complete. 20 / 21

52 THANK YOU! 21 / 21