This post is one of a series which aims to survey the known results on the class of monopolar graphs. If I have missed something, or if you are interested in seeing the proof of an unpublished result, please let me know at rchurchl@uvic.ca.

A graph G is monopolar if its vertices can be partitioned into A and B such that A induces an independent set and B induces a disjoint union of cliques. Such a partition is called a monopolar partition.

Special cases

• Bipartite graphs, which can be partitioned into two independent sets
• Split graphs, which can be partitioned into an independent set and a single clique

Inclusions

• Polar graphs, which can be partitioned into a complete multipartite graph and a disjoint union of cliques

Class properties

• Induced-heredity: any induced subgraph of a monopolar graph is monopolar.
• Additivity: the disjoint union of two monopolar graphs is monopolar.

Graph properties

• If G is a monopolar graph with maximum clique size ω(G), then its chromatic number χ(G) is either ω(G) or ω(G)+1.
• A monopolar graph with no isolated vertices has at most |E(G)| maximal cliques.

  Each maximal clique is either one of the disjoint cliques of B or consists of a single vertex from A and all of its neighbours from one of these cliques. A maximal clique of the first type can be represented by one of its edges, while maximal cliques of the second type can be represented by edges from A to B.

The stable transversal problem for a fixed graph H asks whether a graph contains a stable set that meets every induced copy of H in the graph. Stable transversal problems generalize several vertex partition problems and have been studied for various classes of graphs. Following a result of Farrugia, the stable transversal problem for each C _ℓ with ℓ ≥ 3 is NP-complete. In this paper, we study an ‘edge version’ of these problems. Specifically, we investigate the problem of determining whether a graph contains a matching that meets every copy of H. We show that the problem for C ₃ is polynomial and for each C _ℓwith ℓ ≥ 4 is NP-complete. Our results imply that the stable transversal problem for each C _ℓ with ℓ ≥ 4 remains NP-complete when it is restricted to line graphs. We show by contrast that the stable transversal problem for C ₃, when restricted to line graphs, is polynomial.

This paper, which I presented at IWOCA 2011, gives the first results of what would become my master’s thesis. I am grateful to NSERC for funding this research with a Alexander Graham Bell Canada Graduate Scholarship.

[Photo taken from Robb's memorial page on the BCCUPMS website]

Dr Robb Fry, one of my professors from my Thompson Rivers University days, passed away earlier this year at far too young an age. Robb was a real character, a great teacher, and a lot of fun to know (the Robb Fry Appreciation Group on Facebook had dozens of members from multiple universities even before he died).

I took my second course in linear algebra with Robb, and it was one of the most entertaining courses of my first two years. My course notes – the only full set of notes I ever took in undergrad – were naturally interspersed with records of Robb’s regular witty remarks and quotable quotes. When visiting my parents over the holidays, I dug out these notes, so I thought I’d share some of the memorable episodes from my time with him.

ROBB: Can I call this function  ”f twiddle“ and this one  ”f twiddle twiddle”?
CLASS: Sure.
ROBB: So take “f twiddle twiddle little star…”

(Introducing the notation ∃!)
ROBB: It means “there exists a unique…” but I always read it like it’s a William Shatner thing. THERE EXISTS!

ROBB: You’re defining this… it doesn’t come from God. Especially since God doesn’t exist.
CLASS: Ooooooooooooh
ROBB: Oooh, he has an opinion!

ROBB: An oval…
CLASS: Don’t you mean an ellipse?
ROBB: Yeah, whatever the real term for that is.

ROBB: I’m going to avoid yellow chalk today, because I have a suspicion that one day they’ll find that the stuff that makes it yellow is toxic. That’s going to be someone’s Ph.D. thesis one day, The Toxic Effects of Yellow Chalk, and I don’t want to be part of the study group.

(After introducing the kernel as “the set of all stuff that gets killed by a map”)
ROBB: The best bumper sticker I’ve ever seen had a picture of the colonel from KFC with “I am dead.”

ROBB: If you’re ever reading a paper, and they say they have to prove a technical lemma, brace yourself for some HORRIFIC MATH.

(Talking about stable/invariant sets)
ROBB: I use “invariant” instead of “stable.” A stable set sounds like something for horses. I like horses, mind you, but they shouldn’t be confused with mathematics.

You taught me, inspired me, and motivated me to continue on the path to becoming a mathematician. But more importantly, it was a lot of fun to know you. Thanks, Robb.

The list monopolar partition problem asks whether a graph G together with lists L(v)⊆{0,1},v∈V(G) admits a mapping f:V(G)→{0,1} such that f(v)∈L(v) for each  induces an independent set and f⁻¹(1)induces a disjoint union of cliques in G. This problem is NP-complete in general. We show that the problem is solvable in time O(n²m) for claw-free graphs where n and m are the numbers of vertices and edges respectively in the input graph.

Shortly after solving the monopolar partition problem for line graphs, Jing and I realized that our solution could be used to solve the “precoloured” version of the problem, and then further extended to claw-free graphs. Jing presented this paper at the 8th French Combinatorial Conference in summer 2010.

A graph is polar if its vertex set can be partitioned into A and B in such a way that A induces a complete multipartite graph and B induces a disjoint union of cliques (i.e., the complement of a complete multipartite graph). Polar graphs naturally generalize several classes of graphs such as bipartite, cobipartite, and split graphs. The problem of recognizing polar graphs is NP-complete in general. However, it has been shown to be polynomial for several classes of graphs, including cographs and chordal graphs. In this paper, we study the problem of recognizing graphs whose line graphs are polar. It turns out that the core part of this problem lies in determining whether the edge set of a graph admits a partition (A,B) so that A induces a P₃-free subgraph (i.e., a matching) and B induces a P₄-free subgraph. We give a structural characterization of such graphs. The characterization enables us to devise an O(n) time algorithm to solve the stated recognition problem.

This paper is the result of the research term I took as an undergraduate in the summer of 2009. I am grateful to NSERC for funding my work with a Undergraduate Student Research Award, and to my supervisor and coauthor Jing Huang.

Two weeks after hosting CanaDAM, UVic was also the home of this year’s International Workshop on Combinatorial Algorithms; there I presented a paper by myself, Jing Huang, and Xuding Zhu on cycle-transverse matchings. In the paper, we consider the C_k transverse matching problem: given a graph G, is there a matching M such that G-M contains no cycles of length k? We gave a polynomial-time algorithm for the case k=3, but showed that the problem is NP-complete when k ≥ 4. In fact, when k ≥ 4 is even, the problem is still NP-complete for bipartite graphs.

The case k = 4 has connections with the tatami tiling problem, as Frank Ruskey pointed out to us.

Click here to see the slides of the talk. (PDF)

University innings

Bowling

Fall of wickets: 1-22 (Rajinder, 1.3 ov), 2-32 (Peter, 2.6 ov), 3-40 (Kevin, retired, 4.4 ov), 4-56 (Raj, 7.3 ov), 5-64 (Prabhat, 8.6 ov), 6-66 (David, 9.3 ov), 7-98 (Ross, 13.3 ov), 8-98 (James, 13.5 ov), 9-104 (Shantanav, retired, 14.3 ov)

Highlights: Kevin opened our batting with a massive six into the adjacent pasture, bothering some sheep and losing the game ball. He eventually retired on 25* with an impressive strike rate of 227, thanks to a four and another long six. At the other end of the innings, Shantanav closed with three boundaries and a professional 26*. On a personal note, I hit my first-ever boundary on the way to scoring a season-high 8 runs.

Colts innings

Bowling

Fall of wickets: 1-11 (Ram, 2.1 ov), 2-11 (Trevor, 2.2 ov), 3-15 (Theo, 4.1 ov), 4-15 (Trenton, 4.4 ov), 5-25 (Brett, 5.6 ov), 6-29 (Riley, 6.5 ov), 7-31 (YJ, 7.2 ov), 8-31 (Vatsal, 7.4 ov)

Highlights: After our innings, the clouds closed in further; with raindrops suspended in the air, we had to finish the Colts quickly to fend off a draw. Our confident bowlers did the job in less than 8 overs: David taking four wickets, Kevin one, and Raj three. Brett, a new University recruit playing this match for the Colts, put his baseball experience to work with a boundary in his debut innings to score a total of six.

RESULT: University 104-(7+2) beat Colts 31-8 by 73 runs.

This year, I had the pleasure of attending (and helping organize) CanaDAM, which was held at UVic. I gave a talk in the Structured Graphs and Algorithms session, titled “Partitioning graphs via edge-coloured homomorphisms”. In it, I announced some recently-submitted results on the monopolar partition problem and related problems. The main result unifies all currently known polynomial-time monopolarity results: it reduces the problem to 2SAT for a very large graph class, which contains

• claw-free graphs
• chordal graphs
• co-comparability graphs
• all graphs with no induced cycle of length ≥ 5

A similar technique can be used to solve the unipolar partition problem – which asks whether a (general) graph can be partitioned into a clique and a disjoint union of cliques – in polynomial time.

Click here to see the slides of this talk.

University innings

Bowling

Fall of wickets: 1-6 (Shelby, 0.5 ov), 2-48 (Yaju, 4.6 ov), 3-55 (Rajinder, retired, 5.3 ov), 4-62 (Shantanav, 6.5 ov), 5-112 (Doug, retired, 11.1 ov), 6-112 (Miles, 11.4 ov), 7-119 (Zain, retired, 12.3 ov)

Highlights: We had plenty of boundaries this match from the fast field and slower bowling. Rajinder, Doug, and Zain all retired after making their quarter-centuries. Kim struck back with some first-class bowling — taking one wicket while conceding only a four in her first two overs — but we were still able to set a good target of 140.

Wicket Maidens innings

Bowling

Fall of wickets: 1-13 (Laura, 4.2 ov), 2-17 (Shveta, 5.4 ov), 3-33 (Kim, 9.5 ov), 4-49 (Jenny, 15.4 ov)

Highlights: With some highly economical bowling from Prabhat, Rajinder, and Shelby, we managed to hold the Wicket Maidens to 51 runs, twenty short of their target.

RESULT: University 140-(4+3) beat Wicket Maidens 102-4 by 38 runs.

Metchosin innings

Bowling

Fall of wickets: 1-41 (Ty, 4.3 ov), 2-58 (Jordan, 7.6 ov), 3-75 (Jade, 9.6 ov), 4-87 (Saul, 12.4 ov), 5-96 (Rick, retired, 13.4 ov), 6-108 (Blair, 14.4 ov)

University innings

Bowling

Fall of wickets: 1-10 (Yaju, 2.2 ov), 2-23 (Manni, 4.6 ov), 3-23 (Kevin, 5.2 ov), 4-23 (Peter, 5.3 ov), 5-62 (Shantanav, retired, 9.5 ov), 6-65 (Prabhat, 10.4 ov), 7-71 (Ram, 12.5 ov), 8-73 (Miles, 13.1 ov)

RESULT: Metchosin 113-(5+1) beat University 88-(7+1) by 25 runs.